Parametric Equations
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1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).
x=7t and y=17t, where 0 ≤ t ≤ 1
x=7t and y=17t, where 0 ≤ t ≤ 1
2. Find parametric equations describing the line segment from
![{\displaystyle P(x_{1},y_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c40f1c73dac610d345c535db1cc111415a14d97c)
to
![{\displaystyle Q(x_{2},y_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff3be1197a61155a6be18e775a94dd250bd4cf9)
.
![{\displaystyle x=x_{1}+(x_{2}-x_{1})t{\mbox{ and }}y=y_{1}+(y_{2}-y_{1})t,{\mbox{ where }}0\leq t\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/566149b381b475c92d014bb9ef9e74af723a1164)
![{\displaystyle x=x_{1}+(x_{2}-x_{1})t{\mbox{ and }}y=y_{1}+(y_{2}-y_{1})t,{\mbox{ where }}0\leq t\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/566149b381b475c92d014bb9ef9e74af723a1164)
3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.
![{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7a525a585a07d2f091f2f8d0b90e6dadcbe24)
![{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29d7a525a585a07d2f091f2f8d0b90e6dadcbe24)
Sketch the following polar curves without using a computer.
22.
![{\displaystyle r=2-2\sin(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89eb7d1cea850b34322e26aeb873f19dd1cf5dd6)
23.
![{\displaystyle r^{2}=4\cos(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd40021e28f81795d0ed3d91ded3c59eb9b8c95)
24.
![{\displaystyle r=2\sin(5\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47703934f0d67346a46462299eb7e32cea6b4c06)
Sketch the following sets of points.
25.
![{\displaystyle \{(r,\theta ):\theta =2\pi /3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1836249e0f007444d179f372e524c17be93855b2)
26.
![{\displaystyle \{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2249db3f3e74fd825ddd90e811e091f5d8748c4b)
Calculus in Polar Coordinates
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Find points where the following curves have vertical or horizontal tangents.
40.
![{\displaystyle r=4\cos(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ee4e5618c074debbd695aa5915ad54d2f8144d)
Horizontal tangents occur at points where
. This condition is equivalent to
.
Vertical tangents occur at points where
. This condition is equivalent to
.
The condition for a horizontal tangent gives:
Horizontal tangents occur at
which correspond to the Cartesian points
and
.
The condition for a vertical tangent gives:
Vertical tangents occur at
which correspond to the Cartesian points
and
.
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
Horizontal tangents occur at points where
. This condition is equivalent to
.
Vertical tangents occur at points where
. This condition is equivalent to
.
The condition for a horizontal tangent gives:
Horizontal tangents occur at
which correspond to the Cartesian points
and
.
The condition for a vertical tangent gives:
Vertical tangents occur at
which correspond to the Cartesian points
and
.
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
41.
![{\displaystyle r=2+2\sin(\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b267ee749c4bf29cf381fa49b60e942eaba24807)
Horizontal tangents occur at points where
. This condition is equivalent to
.
Vertical tangents occur at points where
. This condition is equivalent to
.
The condition for a horizontal tangent gives:
Horizontal tangents occur at
which correspond to the Cartesian points
,
,
, and
. Point
corresponds to a vertical cusp however and should be excluded leaving
,
, and
.
The condition for a vertical tangent gives:
Vertical tangents occur at
which correspond to the Cartesian points
,
, and
.
Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/2) Horizontal tangents occur at points where
. This condition is equivalent to
.
Vertical tangents occur at points where
. This condition is equivalent to
.
The condition for a horizontal tangent gives:
Horizontal tangents occur at
which correspond to the Cartesian points
,
,
, and
. Point
corresponds to a vertical cusp however and should be excluded leaving
,
, and
.
The condition for a vertical tangent gives:
Vertical tangents occur at
which correspond to the Cartesian points
,
, and
.
Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/2)
Sketch the region and find its area.
43. The region inside the petals of the rose
![{\displaystyle 4\cos(2\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0840648c13a56333b0db84294f5b07dd0cbd5d3)
and outside the circle
![{\displaystyle r=2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19b110d7eb52a69381b88554e63c8a2aef376c7)
Vectors and Dot Product
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60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)
The general equation for a sphere is
where
is the location of the sphere's center and
is the sphere's radius.
It is already known that the sphere's center is
. The sphere's radius is the distance between (1,2,0) and (3,4,5) which is
.
Therefore the sphere's equation is:
. The general equation for a sphere is
where
is the location of the sphere's center and
is the sphere's radius.
It is already known that the sphere's center is
. The sphere's radius is the distance between (1,2,0) and (3,4,5) which is
.
Therefore the sphere's equation is:
.
61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
62. Find the value of
![{\displaystyle |\mathbf {u} +3\mathbf {v} |}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ff89d47db32f016f21b3e7c6702e825dea4941)
if
![{\displaystyle \mathbf {u} =\langle 1,3,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/477838fad43d0f1b62ace6ca15646686ae58da6d)
and
![{\displaystyle \mathbf {v} =\langle 3,0,2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bb0b511446ea75a6c8f3c69694028f87e6514a)
.
Therefore:
.
![{\displaystyle {\sqrt {145}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ce8cd2e34de1c8c186bd803d059665b0a30215)
.
Therefore:
.
![{\displaystyle {\sqrt {145}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ce8cd2e34de1c8c186bd803d059665b0a30215)
63. Find all unit vectors parallel to
![{\displaystyle \langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7035fe004039b3f11a88be5758a290c80f74315)
The length of
is
. Therefore
is a unit vector that points in the same direction as
, and
is a unit vector that points in the opposite direction as
.
are the unit vectors that are parallel to
. The length of
is
. Therefore
is a unit vector that points in the same direction as
, and
is a unit vector that points in the opposite direction as
.
are the unit vectors that are parallel to
.
64. Prove one of the distributive properties for vectors in
![{\displaystyle \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
:
![{\displaystyle c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/864a42744bed5d422284ec8c5b6b441afd443543)
.
.
65. Find all unit vectors orthogonal to
![{\displaystyle 3\mathbf {i} +4\mathbf {j} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7980e9e0edd2c5642a4ca348fc246c1a368b6aec)
in
![{\displaystyle \mathbb {R} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd)
66. Find all unit vectors orthogonal to
![{\displaystyle 3\mathbf {i} +4\mathbf {j} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7980e9e0edd2c5642a4ca348fc246c1a368b6aec)
in
![{\displaystyle \mathbb {R} ^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5)
All vectors
that are orthogonal to
must satisfy
.
The set of possible values of
is
. The restriction that
becomes
.
The set of possible
and
is an ellipse with radii
and
. One possible parameterization of
and
is
and
where
. This parameterization yields
where
as the complete set of unit vectors that are orthogonal to
.
Re-parameterizing by letting
gives the set ![{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,\pm {\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/087456257f4b52c3bf13601c2a02680a406617d4)
All vectors
that are orthogonal to
must satisfy
.
The set of possible values of
is
. The restriction that
becomes
.
The set of possible
and
is an ellipse with radii
and
. One possible parameterization of
and
is
and
where
. This parameterization yields
where
as the complete set of unit vectors that are orthogonal to
.
Re-parameterizing by letting
gives the set ![{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,\pm {\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/087456257f4b52c3bf13601c2a02680a406617d4)
67. Find all unit vectors that make an angle of
![{\displaystyle \pi /3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56c1a0cd8279cea58b0ccb583e75a0ee93975883)
with the vector
![{\displaystyle \langle 1,2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a88f11e3a0bed44340579f2879872f846997685)
The angle that
makes with the x-axis is
counterclockwise.
Making a both a clockwise and a counterclockwise rotation of
gives
![{\displaystyle ={\frac {\sqrt {5}}{10}}\left\langle 1\pm 2{\sqrt {3}},\ 2\mp {\sqrt {3}}\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c6af9b8bb4d7b7c7ce9022179ffe2cfd4b5c7c)
The angle that
makes with the x-axis is
counterclockwise.
Making a both a clockwise and a counterclockwise rotation of
gives
![{\displaystyle ={\frac {\sqrt {5}}{10}}\left\langle 1\pm 2{\sqrt {3}},\ 2\mp {\sqrt {3}}\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c6af9b8bb4d7b7c7ce9022179ffe2cfd4b5c7c)
Find
and
80.
![{\displaystyle \mathbf {u} =\langle -4,1,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cc81e26260bf95572ed05c035c466d72d7fca5)
and
![{\displaystyle \mathbf {v} =\langle 0,1,-1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/446af0e93cd08418eab3e6d072ae86473aadcd16)
![{\displaystyle \mathbf {v} \times \mathbf {u} =-(\mathbf {u} \times \mathbf {v} )=\langle 2,4,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f6c7c335895bc043fb05ffd880b18b4329cd23)
![{\displaystyle \mathbf {v} \times \mathbf {u} =-(\mathbf {u} \times \mathbf {v} )=\langle 2,4,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f6c7c335895bc043fb05ffd880b18b4329cd23)
81.
![{\displaystyle \mathbf {u} =\langle 1,2,-1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6ea14a685318f6b12f349ce3128623b1199278a)
and
![{\displaystyle \mathbf {v} =\langle 3,-4,6\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f28ec1e044d6b3b86b42f67e33a062be68e8cee)
![{\displaystyle \mathbf {v} \times \mathbf {u} =-(\mathbf {u} \times \mathbf {v} )=\langle -8,9,10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a744f0bb2d330c52a32a433a52b7d5c2cb05eb)
![{\displaystyle \mathbf {v} \times \mathbf {u} =-(\mathbf {u} \times \mathbf {v} )=\langle -8,9,10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a744f0bb2d330c52a32a433a52b7d5c2cb05eb)
Find the area of the parallelogram with sides
and
.
82.
![{\displaystyle \mathbf {u} =\langle -3,0,2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4791ab7f516d8d8fc3ed30da7188e2e123e78b30)
and
![{\displaystyle \mathbf {v} =\langle 1,1,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/20c8ec0469b1d3ac62fdf5354710dca43bd56431)
83.
![{\displaystyle \mathbf {u} =\langle 8,2,-3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4fac47a86b5eba27cab113cae9a04ce5e629cb)
and
![{\displaystyle \mathbf {v} =\langle 2,4,-4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9f91dca4fd3f4b64bdeb5cee7d3bb0097db379)
84. Find all vectors that satisfy the equation
![{\displaystyle \langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a5ab6d6ab73a6ad45be8cbcd11b27c164a49186)
The cross product is orthogonal to both multiplicand vectors.
should be orthogonal to both
and
. However,
so
and
are not orthogonal. The equation
is never true, and therefore the set of vectors
that satisfy the equation is
"None".
The cross product is orthogonal to both multiplicand vectors.
should be orthogonal to both
and
. However,
so
and
are not orthogonal. The equation
is never true, and therefore the set of vectors
that satisfy the equation is
"None".
85. Find the volume of the parallelepiped with edges given by position vectors
![{\displaystyle \langle 5,0,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/890ffd433a0f58e918c9acffb407ce90010bf91e)
,
![{\displaystyle \langle 1,4,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7340622045dacc04f3ea01ee2afd99c50730fadb)
, and
![{\displaystyle \langle 2,2,7\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcbb9a7383295fdb038fa6a4d859a2ffca42dbed)
The volume of a parallelepiped with edges defined by the vectors
,
, and
is the absolute value of the scalar triple product:
.
The volume of a parallelepiped with edges defined by the vectors
,
, and
is the absolute value of the scalar triple product:
.
86. A wrench has a pivot at the origin and extends along the
x-axis. Find the magnitude and the direction of the torque at the pivot when the force
![{\displaystyle \mathbf {F} =\langle 1,2,3\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae64933db3c226c9af2a7843fd90207bab5b186e)
is applied to the wrench
n units away from the origin.
The moment arm is
, so the torque applied is
The magnitude of the torque is
. The torque's direction is
. The moment arm is
, so the torque applied is
The magnitude of the torque is
. The torque's direction is
.
Prove the following identities or show them false by giving a counterexample.
Calculus of Vector-Valued Functions
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100. Differentiate
![{\displaystyle \mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/17314a7c81efe26c5e82f9a1b648eecc13b143b4)
.
![{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,\cos(t)-t\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d62b69912aaa199d91c90edfa813170329571a)
![{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,\cos(t)-t\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/46d62b69912aaa199d91c90edfa813170329571a)
101. Find a tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d58da880ed1d898cf6293bac492a86129fc61e95)
at the point
![{\displaystyle t=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/970dea4a5f5ec5355c4cdd62f6396fbc8b1baaa1)
.
so a possible a tangent vector at
is ![{\displaystyle {\dot {\mathbf {r} }}(1)=\langle 8,9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/67b9252dfb26722105eb357d347265678c02a123)
so a possible a tangent vector at
is ![{\displaystyle {\dot {\mathbf {r} }}(1)=\langle 8,9,-10\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/67b9252dfb26722105eb357d347265678c02a123)
102. Find the unit tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e23189681f86b836c11ef60d9c34e76366950a)
.
so the unit tangent vector is ![{\displaystyle \mathbf {T} (t)={\frac {{\dot {\mathbf {r} }}(t)}{|{\dot {\mathbf {r} }}(t)|}}={\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9013230fe789af143dce695b984f9aebcb763c)
so the unit tangent vector is ![{\displaystyle \mathbf {T} (t)={\frac {{\dot {\mathbf {r} }}(t)}{|{\dot {\mathbf {r} }}(t)|}}={\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9013230fe789af143dce695b984f9aebcb763c)
103. Find the unit tangent vector for the curve
![{\displaystyle \mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f30552af99a8f55083220a2960a8a6697bad79)
at the point
![{\displaystyle t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c)
.
so the unit tangent vector is
At
: ![{\displaystyle \mathbf {T} (0)={\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f576fb01ab74868bd925d30b69066411211fdad7)
so the unit tangent vector is
At
: ![{\displaystyle \mathbf {T} (0)={\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f576fb01ab74868bd925d30b69066411211fdad7)
104. Find
![{\displaystyle \mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1)
if
![{\displaystyle \mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c315146fae77e7ef4859b40635eb80771161efb9)
and
![{\displaystyle \mathbf {r} (1)=\langle 2,3,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62fedc15f45ae9d0880817374bf1cd8914d3f99)
.
For an arbitrary
the position
can be computed by the integral
.
![{\displaystyle =\left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln(t)+4\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9bfc92e4ca4800bcc510775bce5b6827ef3bd6)
For an arbitrary
the position
can be computed by the integral
.
![{\displaystyle =\left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln(t)+4\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9bfc92e4ca4800bcc510775bce5b6827ef3bd6)
105. Evaluate
![{\displaystyle \displaystyle \int _{0}^{\ln 2}(e^{-t}\mathbf {i} +2e^{2t}\mathbf {j} -4e^{t}\mathbf {k} )dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5efb598fa2b457d66dc7fa4b71cd26ce1d9de1)
![{\displaystyle =\langle 1/2,3,-4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/128750e2dfb9b15e3ba474dc5baa7a155f7f3699)
![{\displaystyle =\langle 1/2,3,-4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/128750e2dfb9b15e3ba474dc5baa7a155f7f3699)
120. Find velocity, speed, and acceleration of an object if the position is given by
![{\displaystyle \mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a4e8d4f758878d2c7cf81dd09d71840a47f59e)
.
,
, ![{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d6ed7acbb6709b935b4873a4e3e28ca912d7c6)
,
, ![{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6d6ed7acbb6709b935b4873a4e3e28ca912d7c6)
121. Find the velocity and the position vectors for
![{\displaystyle t\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060)
if the acceleration is given by
![{\displaystyle \mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60d29ee4e6618c95f5ecdca838474e8f6c0d2607)
.
![{\displaystyle =\langle e^{-t}+2t-1,t^{2}/2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/377dcc6ec9ecad63cac0a7706c3068b798c2d9f2)
![{\displaystyle =\langle e^{-t}+2t-1,t^{2}/2\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/377dcc6ec9ecad63cac0a7706c3068b798c2d9f2)
Find the length of the following curves.
140.
![{\displaystyle \mathbf {r} (t)=\langle 4\cos(3t),4\sin(3t)\rangle ,\ t\in [0,2\pi /3].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3681f0d46ab5bdb85098ee74c0058797105aace)
For an infinitesimal step
, the length traversed is approximately
.
The total length is therefore:
![{\displaystyle L=\int _{t=0}^{2\pi /3}ds=\int _{t=0}^{2\pi /3}12dt=12t{\Bigg |}_{t=0}^{2\pi /3}=8\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/57adb679c6344e7cc7c22d38f9d7bc786c4b8bd9)
For an infinitesimal step
, the length traversed is approximately
.
The total length is therefore:
![{\displaystyle L=\int _{t=0}^{2\pi /3}ds=\int _{t=0}^{2\pi /3}12dt=12t{\Bigg |}_{t=0}^{2\pi /3}=8\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/57adb679c6344e7cc7c22d38f9d7bc786c4b8bd9)
141.
![{\displaystyle \mathbf {r} (t)=\langle 2+3t,1-4t,3t-4\rangle ,\ t\in [1,6].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1513778e4cae9f336be929232f7054e2840ec58f)
For an infinitesimal step
, the length traversed is approximately
.
The total length is therefore:
![{\displaystyle L=\int _{t=1}^{6}ds=\int _{t=1}^{6}{\sqrt {34}}\cdot dt={\sqrt {34}}\cdot t{\Bigg |}_{t=1}^{6}=5{\sqrt {34}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcd260adbcfb1f6c1db0cc13ddc3fb6b42c797d)
For an infinitesimal step
, the length traversed is approximately
.
The total length is therefore:
![{\displaystyle L=\int _{t=1}^{6}ds=\int _{t=1}^{6}{\sqrt {34}}\cdot dt={\sqrt {34}}\cdot t{\Bigg |}_{t=1}^{6}=5{\sqrt {34}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3bcd260adbcfb1f6c1db0cc13ddc3fb6b42c797d)
Parametrization and Normal Vectors
edit
142. Find a description of the curve that uses arc length as a parameter:
![{\displaystyle \mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9715ec537d31fdc34e2ac352df9235bfa517bc28)
For an infinitesimal step
, the length traversed is approximately
Given an upper bound of
, the arc length swept out from
to
is:
The arc length spans a range from
to
. For an arc length of
, the upper bound on
that generates an arc length of
is
, and the point at which this upper bound occurs is: ![{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e99391b12706ea35251d2bfb9592712b75ee995f)
For an infinitesimal step
, the length traversed is approximately
Given an upper bound of
, the arc length swept out from
to
is:
The arc length spans a range from
to
. For an arc length of
, the upper bound on
that generates an arc length of
is
, and the point at which this upper bound occurs is: ![{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e99391b12706ea35251d2bfb9592712b75ee995f)
143. Find the unit tangent vector
T and the principal unit normal vector
N for the curve
![{\displaystyle \mathbf {r} (t)=\langle t^{2},t\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3689f66f5ffe1373715b1c288720ee58e03bec63)
Check that
T⋅
N=0.
A tangent vector is
. Normalizing this vector to get the unit tangent vector gives:
A vector that has the direction of the principal unit normal vector is
Normalizing
gives the principal unit normal vector:
![{\displaystyle =\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5b1a4b6b81fa5d3d60644b82c561da565d6379)
A tangent vector is
. Normalizing this vector to get the unit tangent vector gives:
A vector that has the direction of the principal unit normal vector is
Normalizing
gives the principal unit normal vector:
![{\displaystyle =\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5b1a4b6b81fa5d3d60644b82c561da565d6379)
Equations of Lines And Planes
edit
160. Find an equation of a plane passing through points
![{\displaystyle (1,1,2),\ (1,2,2),\ (-1,0,1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0521801e98c7130ffc5dc1e6682f8ed4e81d564d)
Let
denote a plane that contains points
,
, and
. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
.
The displacement from
to
, which is
, and the displacement from
to
, which is
, are both contained by
so the cross product of these two displacements forms a candidate
:
Any of
,
, and
is a candidate
. Let
The equation becomes
![{\displaystyle \iff x-2z+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28506f31d900f4f4851d2f703b32f631a1d151c3)
Let
denote a plane that contains points
,
, and
. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
.
The displacement from
to
, which is
, and the displacement from
to
, which is
, are both contained by
so the cross product of these two displacements forms a candidate
:
Any of
,
, and
is a candidate
. Let
The equation becomes
![{\displaystyle \iff x-2z+3=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28506f31d900f4f4851d2f703b32f631a1d151c3)
161. Find an equation of a plane parallel to the plane 2x−y+z=1 passing through the point (0,2,-2)
Let
denote a plane that is parallel to the plane
and contains the point
. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
.
Any vector
that is orthogonal to
is also orthogonal to
and vice versa. Since
, the coefficient vector
is orthogonal to
, so a candidate
is
.
Since point
is contained by
, let
.
The equation becomes
![{\displaystyle \iff 2x-y+z+4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f94eb5c0cba11ae9fbe1628fea13f782ecfd8b)
Let
denote a plane that is parallel to the plane
and contains the point
. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
.
Any vector
that is orthogonal to
is also orthogonal to
and vice versa. Since
, the coefficient vector
is orthogonal to
, so a candidate
is
.
Since point
is contained by
, let
.
The equation becomes
![{\displaystyle \iff 2x-y+z+4=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17f94eb5c0cba11ae9fbe1628fea13f782ecfd8b)
162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).
Let
denote an arbitrary plane. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
. Therefore the equation that defines
is
.
The equation
is equivalent to
. This implies that the coefficient vector
is orthogonal to the plane defined by
. A line that passes through point
and is parallel to
is parameterized by:
![{\displaystyle \mathbf {r} (t)=\langle 5,5,5\rangle +t\langle 1,1,2\rangle =\langle 5+t,5+t,5+2t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5447c49a1abe7f66a2e06f678ef717b41d200911)
Let
denote an arbitrary plane. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
. Therefore the equation that defines
is
.
The equation
is equivalent to
. This implies that the coefficient vector
is orthogonal to the plane defined by
. A line that passes through point
and is parallel to
is parameterized by:
![{\displaystyle \mathbf {r} (t)=\langle 5,5,5\rangle +t\langle 1,1,2\rangle =\langle 5+t,5+t,5+2t\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5447c49a1abe7f66a2e06f678ef717b41d200911)
163. Find an equation of the line where planes x+2y−z=1 and x+y+z=1 intersect.
164. Find the angle between the planes x+2y−z=1 and x+y+z=1.
Let
denote an arbitrary plane. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
. Therefore the equation that defines
is
.
Let
be the plane described by
and
be the plane described by
Since
, the coefficient vector
is orthogonal to
.
Since
, the coefficient vector
is orthogonal to
.
The angle
between
and
is equivalent to the angle between
and
:
![{\displaystyle =\arccos {\frac {2}{\sqrt {18}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9427a5156cac3bc1bae5da65cd2b82d0100dde3a)
Let
denote an arbitrary plane. Let
denote an arbitrary vector that is orthogonal to
, and
denote the position vector of an arbitrary point contained by
. A point at position vector
is contained by
if and only if the displacement from
is orthogonal to
. This yields the equation
. Therefore the equation that defines
is
.
Let
be the plane described by
and
be the plane described by
Since
, the coefficient vector
is orthogonal to
.
Since
, the coefficient vector
is orthogonal to
.
The angle
between
and
is equivalent to the angle between
and
:
![{\displaystyle =\arccos {\frac {2}{\sqrt {18}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9427a5156cac3bc1bae5da65cd2b82d0100dde3a)
165. Find the distance from the point (3,4,5) to the plane x+y+z=1.
Given a unit length vector
, consider an axis
oriented in the direction of
. The "
coordinate" is determined by orthogonally projecting points onto the
axis. Given a position vector
, the expression
computes the
coordinate.
The equation
is equivalent to
Letting
, the plane
consists of all points whose
coordinate is
. The
coordinate of
is
.
The distance between the plane
and the point
along the
axis is
The distance
is the distance between the point and plane along a direction that is orthogonal to the plane, and is hence the shortest distance. Given a unit length vector
, consider an axis
oriented in the direction of
. The "
coordinate" is determined by orthogonally projecting points onto the
axis. Given a position vector
, the expression
computes the
coordinate.
The equation
is equivalent to
Letting
, the plane
consists of all points whose
coordinate is
. The
coordinate of
is
.
The distance between the plane
and the point
along the
axis is
The distance
is the distance between the point and plane along a direction that is orthogonal to the plane, and is hence the shortest distance.
Limits And Continuity
edit
Evaluate the following limits.
180.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcb684f07cfbcf3df20297f75d0bb3fe0179fb5)
![{\displaystyle =-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2e0de7bce96fe4f51033de551d4c6387869089)
![{\displaystyle =-2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc2e0de7bce96fe4f51033de551d4c6387869089)
181.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c9d03ed5480a6d5c2799367ccb412cdda00bcb6)
![{\displaystyle ={\frac {1}{{\sqrt {4+5}}+3}}=1/6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55efdefd0c972a8390d517d9a5c9f49177d378)
![{\displaystyle ={\frac {1}{{\sqrt {4+5}}+3}}=1/6}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e55efdefd0c972a8390d517d9a5c9f49177d378)
At what points is the function f continuous?
183.
![{\displaystyle f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67eb097c7031711caf5807cb61be6827e4c211b8)
All points (x,y) except for (0,0) and the line y=x+1
All points (x,y) except for (0,0) and the line y=x+1
Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)
184.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0bf345f40643afd561dcdcbc668e1a58f7f0e6b)
The limit is 1 along the line y=x, and −1 along the line y=−x
The limit is 1 along the line y=x, and −1 along the line y=−x
186.
![{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a2e399deb0b86fc6d3c12e420ee28ced53963c)
The limit is 1 along the line y=0, and −1 along the line x=0
The limit is 1 along the line y=0, and −1 along the line x=0
200. Find
![{\displaystyle \partial z/\partial x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc9b120218ae54884244413cd7a4160ca963844)
if
![{\displaystyle \displaystyle z(x,y)={\frac {1}{\ln(xy)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91acfd48a947d8db5fc8b3d91d5b4e5681327453)
201. Find all three partial derivatives of the function
![{\displaystyle \displaystyle f(x,y,z)=xe^{y^{2}+z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2161eebe16dc5f0baa34fbfe03d9e5694ec9c224)
![{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5683300bea5c2e6e1d4779ad4baa4105feafdaf)
![{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5683300bea5c2e6e1d4779ad4baa4105feafdaf)
Find the four second partial derivatives of the following functions.
202.
![{\displaystyle f(x,y)=\cos(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1d272176c97cbb1fbeea4554140a878c69657a)
203.
![{\displaystyle f(x,y)=xe^{y}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6cc1e77a2278e0b5e037b2279dc919f171f264)
Find
220.
![{\displaystyle f(x,y)=x^{2}y-xy^{3},\ x(t)=t^{2},\ y(t)=t^{-2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ed322590bc05b7e895863ed7f0b28dc54379541)
221.
![{\displaystyle f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b75f46b32b6976789d4d1b3dee6c2f38478769b9)
222.
![{\displaystyle \displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1907538ad615ebbb0f65839249d79cec414a5611)
Find
223.
![{\displaystyle f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc83b657165c2155f0231dde30d50a440a613474)
224.
![{\displaystyle \displaystyle f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/283e90dd2ce5331b454967d79bef102af7a37748)
225. The volume of a pyramid with a square base is
![{\displaystyle V={\frac {1}{3}}x^{2}h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9ce6ad029040bf3134c9692d0af40f5b4f2a25)
, where
x is the side of the square base and
h is the height of the pyramid. Suppose that
![{\displaystyle \displaystyle x(t)={\frac {t}{t+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27e535cd1991ddf259ea8b9985f08322c268f825)
and
![{\displaystyle \displaystyle h(t)={\frac {1}{t+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bc7f9494c057e079c6f262a20f84cb7757d04f)
for
![{\displaystyle t\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb260e03f8f2bf09d20a8369fb31b3988501014)
Find
![{\displaystyle V'(t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3002031ba583aa07c59da516f16e188bcd3a5602)
Find an equation of a plane tangent to the given surface at the given point(s).
240.
![{\displaystyle xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/114a1efaa3ea6778f3235a323753f3f335683488)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
.
The point
lies in the surface, and the tangent plane is
.
The tangent planes are therefore:
![{\displaystyle (x-1)+{\frac {1}{2}}(y-2)+{\sqrt {3}}(z-\pi /6)=0,\ (x+1)+{\frac {1}{2}}(y+2)+{\sqrt {3}}(z-5\pi /6)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae0ee90e1005cb3fa520f040d9ef00e908fb2b8)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
.
The point
lies in the surface, and the tangent plane is
.
The tangent planes are therefore:
![{\displaystyle (x-1)+{\frac {1}{2}}(y-2)+{\sqrt {3}}(z-\pi /6)=0,\ (x+1)+{\frac {1}{2}}(y+2)+{\sqrt {3}}(z-5\pi /6)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae0ee90e1005cb3fa520f040d9ef00e908fb2b8)
241.
![{\displaystyle z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a39ab5e2b06a6e0bb1bb5d800d8b353412165f6f)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
.
The point
lies in the surface, and the tangent plane is
.
The tangent planes are therefore:
![{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc74a7f63a4e97b73e815342f8d2247d3839ecc)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
.
The point
lies in the surface, and the tangent plane is
.
The tangent planes are therefore:
![{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc74a7f63a4e97b73e815342f8d2247d3839ecc)
242.
![{\displaystyle z=\tan ^{-1}(x+y),\ (0,0,0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582b81ad602077be76fc77165bde03f297d6cf20)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
![{\displaystyle \iff x+y-z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29061ca58adfcb4afb634da2cfe9a364b0619cb5)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
![{\displaystyle \iff x+y-z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29061ca58adfcb4afb634da2cfe9a364b0619cb5)
243.
![{\displaystyle \sin(xyz)=1/2,\ (\pi ,1,1/6).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16e37b3f7e7699af03c57fa0212d401a6e19fcea)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
![{\displaystyle \iff x+\pi y+6\pi z=3\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/51dc6c4a29f84e468e0774cc31391690969d64a6)
Start with a point
that is on the surface. Perturbing the
,
, and
coordinates by infinitesimal amounts
,
, and
respectively, changes the value of
by the infinitesimal amount
, and the value of
by
. To remain in the surface it must be the case that
.
To linearly extrapolate the condition
to a tangent plane at
, replace the infinitesimal perturbations
,
, and
with large perturbations
,
, and
to get
. Any point in the tangent plane at
can be reached by an appropriate choice of
,
, and
where
. Any point
in the tangent plane at
must satisfy
.
The point
lies in the surface, and the tangent plane is
![{\displaystyle \iff x+\pi y+6\pi z=3\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/51dc6c4a29f84e468e0774cc31391690969d64a6)
Maximum And Minimum Problems
edit
Find critical points of the function f. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.
How to find local minimums, local maximums, and saddle points for a function with an unconstrained domain.
Start with a candidate point
, and envision that the
and
coordinates are changing at rates of
and
respectively:
and
. The rate of change in
is
.
is a local minimum only if
for all
. This occurs iff
.
is a local maximum only if
for all
. This occurs iff
.
Points where
are "critical points" and may contain a local minimum, a local maximum, or a saddle point. It is then needed to classify the critical point.
The second derivative is
.
- A critical point is a local minimum iff
for all
.
- A critical point is a local maximum iff
for all
.
- A critical point that is neither a local minimum nor a local maximum is a saddle point.
While it will not be shown here,
can attain both positive and negative values iff
.
- A critical point is a local minimum if
and ![{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},{\frac {\partial ^{2}f}{\partial y^{2}}}\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286139607cbec04ba3bcd3b886c015d9e4ac845d)
- A critical point is a local maximum if
and ![{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}},{\frac {\partial ^{2}f}{\partial y^{2}}}\leq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5285d901eabaa728b6d66bf0aba5f7027c9e85b2)
- A critical point is a saddle point if
![{\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}{\frac {\partial ^{2}f}{\partial y^{2}}}-\left({\frac {\partial ^{2}f}{\partial y\partial x}}\right)^{2}<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c527e3ac9c095cfa4b4f1f5ab6d45b8a84970413)
The quantity
will be called the "discriminant".
260.
![{\displaystyle f(x,y)=x^{4}+2y^{2}-4xy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a1c445e7214a6b2d924712c498a15a07d6dba2)
The first order derivatives are:
and
. Finding the critical points is done by solving the equations
. Substituting
in place of
in the first equation gives
. This gives the critical points
.
The second order derivatives are
;
; and
. The discriminant is:
.
For critical point
, the discriminant is
and
so
is a local minimum.
For critical point
, the discriminant is
so
is a saddle point.
For critical point
, the discriminant is
and
so
is a local minimum.
Local minima at (1,1) and (−1,−1), saddle at (0,0) The first order derivatives are:
and
. Finding the critical points is done by solving the equations
. Substituting
in place of
in the first equation gives
. This gives the critical points
.
The second order derivatives are
;
; and
. The discriminant is:
.
For critical point
, the discriminant is
and
so
is a local minimum.
For critical point
, the discriminant is
so
is a saddle point.
For critical point
, the discriminant is
and
so
is a local minimum.
Local minima at (1,1) and (−1,−1), saddle at (0,0)
261.
![{\displaystyle f(x,y)=\tan ^{-1}(xy)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e84fbb10c429427caca59ef04d46bb2bdf3607)
The first order derivatives are:
and
. Finding the critical points is done by solving the equations
and
. So the only critical point is
.
The second order derivatives are
;
; and
. The discriminant is
For the critical point
, the discriminant is
so
is a saddle point.
Saddle at (0,0) The first order derivatives are:
and
. Finding the critical points is done by solving the equations
and
. So the only critical point is
.
The second order derivatives are
;
; and
. The discriminant is
For the critical point
, the discriminant is
so
is a saddle point.
Saddle at (0,0)
262.
![{\displaystyle f(x,y)=2xye^{-x^{2}-y^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59076ebe29ea207c890ee9a748b89ba48db22315)
The first order derivatives are:
and
. Finding the critical points is done by solving the equations
. The critical points are
.
The second order derivatives are:
;
; and
. The discriminant is
.
For critical point
, the discriminant is
so
is a saddle point.
For critical points
and
, the discriminant is
and
so
and
are local maximums.
For critical points
and
, the discriminant is
and
so
and
are local minimums.
Saddle at (0,0), local maxima at
local minima at ![{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177e9620f311a421e11d89435014d73605eebddb)
The first order derivatives are:
and
. Finding the critical points is done by solving the equations
. The critical points are
.
The second order derivatives are:
;
; and
. The discriminant is
.
For critical point
, the discriminant is
so
is a saddle point.
For critical points
and
, the discriminant is
and
so
and
are local maximums.
For critical points
and
, the discriminant is
and
so
and
are local minimums.
Saddle at (0,0), local maxima at
local minima at ![{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177e9620f311a421e11d89435014d73605eebddb)
Find absolute maximum and minimum values of the function f on the set R.
How to find candidate points for the absolute minimum and maximum of a function with a constrained domain.
Let
denote the function for which the absolute minimum and maximum is sought. Let the domain be constrained to all points
where
where
is an appropriate function over
.
Start with a candidate point
where
, and envision that the
and
coordinates are changing at rates of
and
respectively:
and
. The rate of change in
is
. Since it is required that
, it must be the case that
.
is a local minimum or maximum only if
for all
where
. This occurs iff the gradient
is parallel to the gradient
. This condition can be quantified by
where factor
is a "Lagrange multiplier".
Points
where
and
for some
are candidate points for the absolute minimum or maximum. If the domain has any corners, then these corners are also candidate points.
263.
![{\displaystyle f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25fa736e06e7208a73dd247973bc3096f95d102b)
Candidate points will be derived from two sources: Critical points of the function
assuming an unconstrained domain, and candidate points assuming the restriction
.
The first order derivatives of
are
and
, so the only critical point where
occurs at
. This critical point lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. If
, then the only restriction left on
is
and
. This gives two candidate points
. If
, then
which is never true. Hence the only valid candidate points derived by restricting the domain to
are
.
and
.
In total, the candidates are
,
, and
.
The absolute minimum of
occurs at
, and the absolute maximum of
occurs at
.
Maximum of 9 at (0,−2) and minimum of 0 at (0,1) Candidate points will be derived from two sources: Critical points of the function
assuming an unconstrained domain, and candidate points assuming the restriction
.
The first order derivatives of
are
and
, so the only critical point where
occurs at
. This critical point lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. If
, then the only restriction left on
is
and
. This gives two candidate points
. If
, then
which is never true. Hence the only valid candidate points derived by restricting the domain to
are
.
and
.
In total, the candidates are
,
, and
.
The absolute minimum of
occurs at
, and the absolute maximum of
occurs at
.
Maximum of 9 at (0,−2) and minimum of 0 at (0,1)
264.
R is a closed triangle with vertices (0,0), (2,0), and (0,2).
Triangle
is defined by the constraints
,
, and
.
Candidate points for the absolute minimum and maximum will be derived from 5 sources:
- Critical points of the function
assuming an unconstrained domain.
- Candidate points assuming the restriction
.
- Candidate points assuming the restriction
.
- Candidate points assuming the restriction
.
- The vertex points
.
The first order derivatives of
are
and
, so the only critical point where
is
. The critical point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. The second equation yields
. Substituting
in place of
in the third equation gives
. Substituting
in place of
in the first equation gives
, which then yields
and
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
Finally, we add the vertices
to the lineup of candidate points.
Evaluating
at each candidate point gives
;
;
;
;
; and
. The absolute minimum of
occurs at
, while the absolute maximum of
occurs at all of
.
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1) Triangle
is defined by the constraints
,
, and
.
Candidate points for the absolute minimum and maximum will be derived from 5 sources:
- Critical points of the function
assuming an unconstrained domain.
- Candidate points assuming the restriction
.
- Candidate points assuming the restriction
.
- Candidate points assuming the restriction
.
- The vertex points
.
The first order derivatives of
are
and
, so the only critical point where
is
. The critical point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. These equations are equivalent to
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
The first order derivatives of
are
and
. Candidate points assuming the restriction
must satisfy
for some
. The second equation yields
. Substituting
in place of
in the third equation gives
. Substituting
in place of
in the first equation gives
, which then yields
and
. This yields the candidate point
. Point
lies in
so it remains a valid candidate.
.
Finally, we add the vertices
to the lineup of candidate points.
Evaluating
at each candidate point gives
;
;
;
;
; and
. The absolute minimum of
occurs at
, while the absolute maximum of
occurs at all of
.
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)
Finding the locations and shortest distances between two surfaces.
Consider two surfaces
and
in 3D space defined by the equations
and
respectively. Given a point
from
and a point
from
, if
and
are the points that minimize the distance between
and
, then it must be the case that the displacement
is perpendicular to both surfaces. The gradient vector
is orthogonal to
, and the gradient vector
is orthogonal to
. The displacement vector must be parallel to both gradient vectors:
for some
and
.
Candidate points for the shortest distance between two surfaces must satisfy the following 8 equations:
for some
and
.
265. Find the point on the plane x−y+z=2 closest to the point (1,1,1).
The plane is defined by the equation
, and a normal vector to the plane is
. The closest point
on the plane to the point
is the point where the displacement
is parallel to
. The following equations must be satisfied:
.
The second equation gives
, and replacing
with
in the third and forth equations gives
and
respectively. In the first equation, replacing
with
and
with
gives
. This gives in turn
and
.
The only candidate point for the closest distance is
, so therefore the point on the plane
that is closest to the point
is
. The plane is defined by the equation
, and a normal vector to the plane is
. The closest point
on the plane to the point
is the point where the displacement
is parallel to
. The following equations must be satisfied:
.
The second equation gives
, and replacing
with
in the third and forth equations gives
and
respectively. In the first equation, replacing
with
and
with
gives
. This gives in turn
and
.
The only candidate point for the closest distance is
, so therefore the point on the plane
that is closest to the point
is
.
266. Find the point on the surface
![{\displaystyle z=x^{2}+y^{2}+10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e85bcb9eddb29c1f0e0f57f9a39dfe6e475a4c30)
closest to the plane
![{\displaystyle x+2y-z=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83871bae0ccffff09b006c5ee522241548a59098)
The surface is defined by the equation
, and the plane is defined by the equation
. Given a point
from the surface and a point
from the plane, these two points are closest to each other only if (but not always if) the displacement vector
is a parallel to the surface normal vector
and the plane normal vector
. There must exist factors
and
such that the following 8 equations hold:
The fifth equation is equivalent to
, and the eighth equation is equivalent to
. Eliminating via substitution
and
in all of the other equations gives:
If
, then
and
, and then
and
are the same point which corresponds to an intersection between the surface and the plane. While it will not be shown here, it is relatively simple to show that the surface and plane fail to intersect. Excluding the possibility that
, the equations
and
together imply that
; and the equations
and
together imply that
. The values
and
give
. Hence the point on the surface
that is closest to the plane
is
.
From
and
, it follows that
and
. In the equation
, eliminating
and
via substitution gives
. Hence
and
. The corresponding closest point on the plane is
.
The closest point on the surface is
. The surface is defined by the equation
, and the plane is defined by the equation
. Given a point
from the surface and a point
from the plane, these two points are closest to each other only if (but not always if) the displacement vector
is a parallel to the surface normal vector
and the plane normal vector
. There must exist factors
and
such that the following 8 equations hold:
The fifth equation is equivalent to
, and the eighth equation is equivalent to
. Eliminating via substitution
and
in all of the other equations gives:
If
, then
and
, and then
and
are the same point which corresponds to an intersection between the surface and the plane. While it will not be shown here, it is relatively simple to show that the surface and plane fail to intersect. Excluding the possibility that
, the equations
and
together imply that
; and the equations
and
together imply that
. The values
and
give
. Hence the point on the surface
that is closest to the plane
is
.
From
and
, it follows that
and
. In the equation
, eliminating
and
via substitution gives
. Hence
and
. The corresponding closest point on the plane is
.
The closest point on the surface is
.
Double Integrals over Rectangular Regions
edit
Evaluate the given integral over the region R.
280.
![{\displaystyle \displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed365e6e7ddc55551e813614af01d9c10cdc9af6)
![{\displaystyle 14/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aba8b23625eb49fef65b0a948730453e5e427b7)
![{\displaystyle 14/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aba8b23625eb49fef65b0a948730453e5e427b7)
281.
![{\displaystyle \displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/348d8c966096be3b39bffd752544bc134cb87182)
![{\displaystyle 1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c)
![{\displaystyle 1/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3cf1ef33695c3d98cb09f01e5700f927ce928c)
282.
![{\displaystyle \displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d6288e4ccdb0134f6d2546fdd60591316f57ca)
![{\displaystyle \ln(5/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ec600eb74181b6432e1121b6534b74634bb1a2)
![{\displaystyle \ln(5/3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76ec600eb74181b6432e1121b6534b74634bb1a2)
Evaluate the given iterated integrals.
Double Integrals over General Regions
edit
Evaluate the following integrals.
300.
R is bounded by
x=0,
y=2
x+1, and
y=5−2
x.
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
![{\displaystyle 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f)
301.
R is in the first quadrant and bounded by
x=0,
![{\displaystyle y=x^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e06a3fb2a27eccbf6e31de4fa547f10784df851)
and
![{\displaystyle y=8-x^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/648c65b0720e93c075f8c6eda6a322737362d694)
![{\displaystyle 152/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93d05b5156a3fbbf4e91b395ed5560eef2eb1f1)
![{\displaystyle 152/3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e93d05b5156a3fbbf4e91b395ed5560eef2eb1f1)
Use double integrals to compute the volume of the given region.
Double Integrals in Polar Coordinates
edit
323. Evaluate
![{\displaystyle \displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21709c9db522102a88cb6b15af60ef48a570bb6a)
if
R is the unit disk centered at the origin.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
In the following exercises, sketching the region of integration may be helpful.
341. Find the volume of the solid in the first octant bounded by the plane 2x+3y+6z=12 and the coordinate planes.
![{\displaystyle 8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5)
![{\displaystyle 8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5)
342. Find the volume of the solid in the first octant bounded by the cylinder
![{\displaystyle z=\sin(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afb4c3adbe65cfc1e0dd92a9d6cb5cf9a95090e3)
for
![{\displaystyle y\in [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0281fdf747a9247ed8a56221ddd07b34c2a50097)
, and the planes
y=
x and
x=0.
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![{\displaystyle \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
344. Rewrite the integral
![{\displaystyle \displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7867366b6580949fa9cd0997759140d0baf564e)
in the order
dydzdx.
![{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c8c7ac2b5595f94dad5e896319604344321768)
![{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49c8c7ac2b5595f94dad5e896319604344321768)
Cylindrical And Spherical Coordinates
edit
361. Find the mass of the solid cylinder
![{\displaystyle D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d5655e4f2397642509c6640400d7087e95cf35)
given the density function
![{\displaystyle \delta (r,\theta ,z)=5e^{-r^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae1f6834555d1dd000077a0627aefbf6d5d234ed)
![{\displaystyle 10\pi (1-e^{-9})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd866b11f9b4e7a51ab5791e303533a363471458)
![{\displaystyle 10\pi (1-e^{-9})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd866b11f9b4e7a51ab5791e303533a363471458)
362. Use a triple integral to find the volume of the region bounded by the plane
z=0 and the hyperboloid
![{\displaystyle z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2458a938a5202f933c5e456ce9a26945661051)
![{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8c4357ba9fef651470f9cfc45c0abca8313973)
![{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8c4357ba9fef651470f9cfc45c0abca8313973)
363. If
D is a unit ball, use a triple integral in spherical coordinates to evaluate
![{\displaystyle \iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV}](https://wikimedia.org/api/rest_v1/media/math/render/svg/223a2baa893cb046b86b90b67460890b83810ff1)
![{\displaystyle \pi /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67)
![{\displaystyle \pi /2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b44e3d874a0b229fded7ffce67a0677dd5b8b67)
364. Find the mass of a solid cone
![{\displaystyle \{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f200079667c56d8b47444f8062308745eb385015)
if the density function is
![{\displaystyle \delta (\rho ,\phi ,\theta )=5-z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9478e110db7bc776f218b7884925de3afa3525a)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
Center of Mass and Centroid
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380. Find the center of mass for three particles located in space at (1,2,3), (0,0,1), and (1,1,0), with masses 2, 1, and 1 respectively.
![{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6332a924ef6c692c318b32da86a2c06ebbadfddb)
![{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6332a924ef6c692c318b32da86a2c06ebbadfddb)
384. Find the centroid of the region in the first quadrant bounded by
![{\displaystyle y=\ln(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a4b4bf3980aaa7227cb1fca33d9864a8bb4b47)
,
![{\displaystyle y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/094f824655138f6b11d96a0da32e7f0716ba6959)
, and
![{\displaystyle x=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba47ab1931fc4886c5da08831962cc141d20655)
.
![{\displaystyle ((e^{2}+1)/4,e/2-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e980929a84e77eb3a1afc014870be9172b909595)
![{\displaystyle ((e^{2}+1)/4,e/2-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e980929a84e77eb3a1afc014870be9172b909595)
385. Find the center of mass for the region
![{\displaystyle \{(x,y)\mid x\in [0,4],y\in [0,2]\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e476dabc9b7a7dfe36cd98372488912496465f7)
, with the density
![{\displaystyle \rho (x,y)=1+x/2.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96618c2fa7107b7b8da2fa03cdbf6642860fecbf)
![{\displaystyle (7/3,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/289e8abbf05e509e7826c279bf13227bffb0f31d)
![{\displaystyle (7/3,1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/289e8abbf05e509e7826c279bf13227bffb0f31d)
386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density
![{\displaystyle \rho (x,y)=1+x+y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d666834cb75e6ac4e82dacb1b584a509ea7d2b)
![{\displaystyle (16/11,16/11)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00fa85c5e62696d2809fe8ea75ba6225c46da1ae)
![{\displaystyle (16/11,16/11)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00fa85c5e62696d2809fe8ea75ba6225c46da1ae)
One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.
402. Find and sketch the gradient field
![{\displaystyle \mathbf {F} =\nabla \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c10e4f746d8ab5226bb1db9dacda8b185a89af)
for the potential function
![{\displaystyle \phi (x,y)=\sin(x)\sin(y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0fcec58b1d9a99ce189585809c8acb4c338280)
for
![{\displaystyle |x|\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1f1f5476d6277e581c15192fe9c5312db24dc1)
and
![{\displaystyle |y|\leq \pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/813925ba0fc80d16f44eaa977f167cf8b2e45b8d)
.
![{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f67b952e709d7c5bdd016b8b1d062cb8f935c01)
![{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f67b952e709d7c5bdd016b8b1d062cb8f935c01)
403. Find the gradient field
![{\displaystyle \mathbf {F} =\nabla \phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c10e4f746d8ab5226bb1db9dacda8b185a89af)
for the potential function
![{\displaystyle \phi (x,y,z)=e^{-z}\sin(x+y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc8e16d40c5f7418c908c1de0dbe618a11d3e20)
![{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0055fc3a8578eb4ba273f863b98306b8f388ca8)
![{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0055fc3a8578eb4ba273f863b98306b8f388ca8)
420. Evaluate
![{\displaystyle \int _{C}(x^{2}+y^{2})ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703c80b06911ba8c668f7a7b65ad254860e19f12)
if
C is the line segment from (0,0) to (5,5)
![{\displaystyle {\frac {250{\sqrt {2}}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba575a2afb84a6aae5c4ad2a926af0cf3b0d2fb0)
![{\displaystyle {\frac {250{\sqrt {2}}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba575a2afb84a6aae5c4ad2a926af0cf3b0d2fb0)
421. Evaluate
![{\displaystyle \int _{C}(x^{2}+y^{2})ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/703c80b06911ba8c668f7a7b65ad254860e19f12)
if
C is the circle of radius 4 centered at the origin
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
![{\displaystyle 128\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f3f1bdb494819c8fbaa95c07293525b9581e74)
422. Evaluate
![{\displaystyle \int _{C}(y-z)ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd22650c4db2c96bb56fc346feec15731fad2f85)
if
C is the helix
![{\displaystyle \mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c08e2103ca08229383eb122b8d4d0b8dba3eac9)
![{\displaystyle -2{\sqrt {10}}\pi ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0854611bc680149cac184dd7252ba9cd61ced7c)
![{\displaystyle -2{\sqrt {10}}\pi ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0854611bc680149cac184dd7252ba9cd61ced7c)
423. Evaluate
![{\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d972b821eeed2f0528814bd37e9a210592b600f1)
if
![{\displaystyle \mathbf {F} =\langle x,y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a961e1a2b386f164460acb5da6152e81216d28d)
and
C is the arc of the parabola
![{\displaystyle \mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/642cd051fef71427b9ff15e0885d8b70e1d1f344)
![{\displaystyle 17/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ab81608c89542f38877067d909f02d008d232)
![{\displaystyle 17/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e2ab81608c89542f38877067d909f02d008d232)
Conservative Vector Fields
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Determine if the following vector fields are conservative on
440.
![{\displaystyle \langle -y,x+y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5396577d51124348a0a78d7b9bf1808009db967)
441.
![{\displaystyle \langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/247696a59731cadb50bde2f7a2ad65b101bb379b)
Determine if the following vector fields are conservative on their respective domains in
When possible, find the potential function.
442.
![{\displaystyle \langle y,x,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95c3f194a95b2fc175ee2a7c37b5d627059dbbfa)
![{\displaystyle \phi (x,y,z)=xy+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9029f2e3b55128f6b9656e68cc3ea1cd560363de)
![{\displaystyle \phi (x,y,z)=xy+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9029f2e3b55128f6b9656e68cc3ea1cd560363de)
443.
![{\displaystyle \langle x^{3},2y,-z^{3}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b81e95e08ec308c8593bf3af6880fe7771e190d6)
![{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be02f5b735e399b4548728942eb0f536f021e4b)
![{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be02f5b735e399b4548728942eb0f536f021e4b)
460. Evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cec48bf9a4e2233fa0b67f33d229f55a4e5de9d8)
over the boundary of the region above
y=0 and below
y=
x(2-
x) in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
461. Evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle 0,x^{2}+y^{2}\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/309e8c3600bf24b333876f9533939b626bc1fb78)
over the unit circle centered at the origin in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
462. Evaluate the flux of the field
![{\displaystyle \mathbf {F} =\langle y,-x\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b344898e27a173253259fb9788bb6ea91f16c4)
over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
482. Find the curl of
![{\displaystyle \langle x^{2}-y^{2},xy,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/201e6047a702995a6baa27da278d38652fcc0a0d)
![{\displaystyle \langle 0,0,3y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac2a763560ae0e692e5e8b336f10463741dea07)
![{\displaystyle \langle 0,0,3y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cac2a763560ae0e692e5e8b336f10463741dea07)
484. Prove that the general rotation field
![{\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2bf675ad6ca689142c4864d3e841979e2da5761)
, where
![{\displaystyle \mathbf {a} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a957216653a9ee0d0133dcefd13fb75e36b8b9d)
is a non-zero constant vector and
![{\displaystyle \mathbf {r} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead22d37bd8c3bd1bede7f57f519eb6845e52eeb)
, has zero divergence, and the curl of
![{\displaystyle \mathbf {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/da18bef8c979f3548bb0d8976f5844012d7b8256)
is
![{\displaystyle 2\mathbf {a} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/47844954e772249023020d9a1ffe5235551b76a1)
.
If
, then
, and then
![{\displaystyle \nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a80883b31c0670c839a9bb9a142c9e8204da46)
If
, then
, and then
![{\displaystyle \nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19a80883b31c0670c839a9bb9a142c9e8204da46)
500. Give a parametric description of the plane
![{\displaystyle 2x-4y+3z=16.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a87717df129166d0da156dd69fe8bfa95afa325c)
![{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c129ca25e6fc4fa0e072e4218dba618f6cadac0)
![{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c129ca25e6fc4fa0e072e4218dba618f6cadac0)
501. Give a parametric description of the hyperboloid
![{\displaystyle z^{2}=1+x^{2}+y^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36ce9240c728e0b1c15d44787db78f723b003b80)
![{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e30ae2582ed79c8841a5ec44ce9a38adb627be5)
![{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e30ae2582ed79c8841a5ec44ce9a38adb627be5)
502. Integrate
![{\displaystyle f(x,y,z)=xy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f27606abbc927475726d0fdce8b33b8b7cce268)
over the portion of the plane
z=2−
x−
y in the first octant.
![{\displaystyle 2/{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb0bea2fe9088875244141adf992b51e8fa12c3)
![{\displaystyle 2/{\sqrt {3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb0bea2fe9088875244141adf992b51e8fa12c3)
504. Find the flux of the field
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
across the surface of the cone
![{\displaystyle z^{2}=x^{2}+y^{2},\ z\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdef9c8592f3f4001cfc49e3df0f914ece0b10df)
with normal vectors pointing in the positive
z direction.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
505. Find the flux of the field
![{\displaystyle \mathbf {F} =\langle -y,z,1\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e33820b226c94ee4cc6192525a12b5c353d225a5)
across the surface
![{\displaystyle y=x^{2},\ z\in [0,4],\ x\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75190cd1fce58f3ae022e60f9e482de598e957cd)
with normal vectors pointing in the positive
y direction.
![{\displaystyle -10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1245b26372e6d9637c98edea699eb2daaaf0b)
![{\displaystyle -10}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1245b26372e6d9637c98edea699eb2daaaf0b)
520. Use a surface integral to evaluate the circulation of the field
![{\displaystyle \mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/439c246c646c331987b4ae83272edca755a0360e)
on the boundary of the plane
![{\displaystyle z=4-x-y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06bee1e94db0c06ce0dc9b5c1229985d620a94ce)
in the first octant.
![{\displaystyle {\frac {-128}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0b4a8ac15057ab7d30e9a0d0dc332bfcdc6903)
![{\displaystyle {\frac {-128}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0b4a8ac15057ab7d30e9a0d0dc332bfcdc6903)
522. Use a line integral to find
where
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the upper half of the ellipsoid
![{\displaystyle {\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7784a34bf21a6187e51888ab1fa67dfc3cd2840)
, and
![{\displaystyle \mathbf {n} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46)
points in the direction of the
z-axis.
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
523. Use a line integral to find
where
![{\displaystyle \mathbf {F} =\langle 2y,-z,x-y-z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b6fa35e54dffb79d7db25a7b07efd7bced82f5b)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the part of the sphere
![{\displaystyle x^{2}+y^{2}+z^{2}=25}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ccdca6176fc4c81546ba8ca563cd1b0826e282)
for
![{\displaystyle 3\leq z\leq 5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c6baa3584177c4b3140a5ec7224b3205258e206)
, and
![{\displaystyle \mathbf {n} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a720c341f39f52fd96028dab83edd34d400be46)
points in the direction of the
z-axis.
![{\displaystyle -32\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/955d27cf435b5b3a32cbcde91325ac0188ac6143)
![{\displaystyle -32\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/955d27cf435b5b3a32cbcde91325ac0188ac6143)
Compute the net outward flux of the given field across the given surface.
540.
![{\displaystyle \mathbf {F} =\langle x,-2y,3z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b48865c18e27292e73f16aef87691e57811f222b)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is a sphere of radius
![{\displaystyle {\sqrt {6}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a857de6bca2591cfad08e4378634825b6be66a01)
centered at the origin.
![{\displaystyle 16{\sqrt {6}}\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56e6edd17d188268c63b773bdee7d3388edb122)
![{\displaystyle 16{\sqrt {6}}\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56e6edd17d188268c63b773bdee7d3388edb122)
542.
![{\displaystyle \mathbf {F} =\langle y+z,x+z,x+y\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f09639d07e2dce9384f8cab5e87e2f8a9287d8)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the cube
![{\displaystyle \{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04cb029fb39e62d41aa1ae0064589be574fbd806)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
543.
![{\displaystyle \mathbf {F} =\langle x,y,z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f902ca1df7a2a32a0451a7fa029cbedcb16b911)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the surface of the region bounded by the paraboloid
![{\displaystyle z=4-x^{2}-y^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20cfd74ab05572b01e6f1909fd54396ce0406f28)
and the
xy-plane.
![{\displaystyle 24\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d591208ffb8cd424b01602f3fa652021a4015314)
![{\displaystyle 24\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d591208ffb8cd424b01602f3fa652021a4015314)
544.
![{\displaystyle \mathbf {F} =\langle z-x,x-y,2y-z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/80caeafe4262109998180fa8966c3baec8bf23b9)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.
![{\displaystyle -224\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7f812d29f8cd9a5298efec1616ac631a890aef)
![{\displaystyle -224\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c7f812d29f8cd9a5298efec1616ac631a890aef)
545.
![{\displaystyle \mathbf {F} =\langle x,2y,3z\rangle }](https://wikimedia.org/api/rest_v1/media/math/render/svg/63f64405ed59d8c593f1c5f58456851cb5a68765)
,
![{\displaystyle S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2)
is the boundary of the region between the cylinders
![{\displaystyle x^{2}+y^{2}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec84b90236512e8d27ff1a8f7707b60b63327de1)
and
![{\displaystyle x^{2}+y^{2}=4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b41e02c83bfa96c6d425137298c311a50eeac17)
and cut off by planes
![{\displaystyle z=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92bfc06485cc90286474b14a516a68d8bfdd7b3)
and
![{\displaystyle z=8}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b37480deb7a0d26a9aab0a0c8b56bd0b55199a3b)
![{\displaystyle 144\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0043fece2cb48fd081c8878c51e4fc93ad8ff94d)
![{\displaystyle 144\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0043fece2cb48fd081c8878c51e4fc93ad8ff94d)