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Parametric Equations
edit
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
P
(
x
1
,
y
1
)
{\displaystyle P(x_{1},y_{1})}
to
Q
(
x
2
,
y
2
)
{\displaystyle Q(x_{2},y_{2})}
.
x
=
x
1
+
(
x
2
−
x
1
)
t
and
y
=
y
1
+
(
y
2
−
y
1
)
t
,
where
0
≤
t
≤
1
{\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}
x
=
x
1
+
(
x
2
−
x
1
)
t
and
y
=
y
1
+
(
y
2
−
y
1
)
t
,
where
0
≤
t
≤
1
{\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}
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.
x
=
3
cos
(
−
t
)
,
y
=
1.5
sin
(
−
t
)
{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}
x
=
3
cos
(
−
t
)
,
y
=
1.5
sin
(
−
t
)
{\displaystyle x=3\cos(-t),\ y=1.5\sin(-t)}
21. Find an equation of the line y =mx +b in polar coordinates.
r
=
b
sin
(
θ
)
−
m
cos
(
θ
)
{\displaystyle r={\frac {b}{\sin(\theta )-m\cos(\theta )}}}
r
=
b
sin
(
θ
)
−
m
cos
(
θ
)
{\displaystyle r={\frac {b}{\sin(\theta )-m\cos(\theta )}}}
Sketch the following polar curves without using a computer.
22.
r
=
2
−
2
sin
(
θ
)
{\displaystyle r=2-2\sin(\theta )}
23.
r
2
=
4
cos
(
θ
)
{\displaystyle r^{2}=4\cos(\theta )}
24.
r
=
2
sin
(
5
θ
)
{\displaystyle r=2\sin(5\theta )}
Sketch the following sets of points.
25.
{
(
r
,
θ
)
:
θ
=
2
π
/
3
}
{\displaystyle \{(r,\theta ):\theta =2\pi /3\}}
26.
{
(
r
,
θ
)
:
|
θ
|
≤
π
/
3
and
|
r
|
<
3
}
{\displaystyle \{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}}
Calculus in Polar Coordinates
edit
Find points where the following curves have vertical or horizontal tangents.
40.
r
=
4
cos
(
θ
)
{\displaystyle r=4\cos(\theta )}
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)
41.
r
=
2
+
2
sin
(
θ
)
{\displaystyle r=2+2\sin(\theta )}
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 /4)
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 /4)
Sketch the region and find its area.
Vectors and Dot Product
edit
60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)
(
x
−
1
)
2
+
(
y
−
2
)
2
+
z
2
=
33
{\displaystyle (x-1)^{2}+(y-2)^{2}+z^{2}=33}
(
x
−
1
)
2
+
(
y
−
2
)
2
+
z
2
=
33
{\displaystyle (x-1)^{2}+(y-2)^{2}+z^{2}=33}
61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
63. Find all unit vectors parallel to
⟨
1
,
2
,
3
⟩
{\displaystyle \langle 1,2,3\rangle }
±
1
14
⟨
1
,
2
,
3
⟩
{\displaystyle \pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle }
±
1
14
⟨
1
,
2
,
3
⟩
{\displaystyle \pm {\frac {1}{\sqrt {14}}}\langle 1,2,3\rangle }
64. Prove one of the distributive properties for vectors in
R
3
{\displaystyle \mathbb {R} ^{3}}
:
c
(
u
+
v
)
=
c
u
+
c
v
{\displaystyle c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v} }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}}
66. Find all unit vectors orthogonal to
3
i
+
4
j
{\displaystyle 3\mathbf {i} +4\mathbf {j} }
in
R
3
{\displaystyle \mathbb {R} ^{3}}
⟨
4
5
c
,
−
3
5
c
,
1
−
c
2
⟩
,
c
∈
[
−
1
,
1
]
{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}
⟨
4
5
c
,
−
3
5
c
,
1
−
c
2
⟩
,
c
∈
[
−
1
,
1
]
{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}
Find
u
×
v
{\displaystyle \mathbf {u} \times \mathbf {v} }
v
×
u
{\displaystyle \mathbf {v} \times \mathbf {u} }
80.
u
=
⟨
−
4
,
1
,
1
⟩
{\displaystyle \mathbf {u} =\langle -4,1,1\rangle }
and
v
=
⟨
0
,
1
,
−
1
⟩
{\displaystyle \mathbf {v} =\langle 0,1,-1\rangle }
u
×
v
=
⟨
−
2
,
−
4
,
−
4
⟩
{\displaystyle \mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle }
u
×
v
=
⟨
−
2
,
−
4
,
−
4
⟩
{\displaystyle \mathbf {u} \times \mathbf {v} =\langle -2,-4,-4\rangle }
81.
u
=
⟨
1
,
2
,
−
1
⟩
{\displaystyle \mathbf {u} =\langle 1,2,-1\rangle }
and
v
=
⟨
3
,
−
4
,
6
⟩
{\displaystyle \mathbf {v} =\langle 3,-4,6\rangle }
u
=
⟨
8
,
−
9
,
−
10
⟩
{\displaystyle \mathbf {u} =\langle 8,-9,-10\rangle }
u
=
⟨
8
,
−
9
,
−
10
⟩
{\displaystyle \mathbf {u} =\langle 8,-9,-10\rangle }
Find the area of the parallelogram with sides
u
{\displaystyle \mathbf {u} }
v
{\displaystyle \mathbf {v} }
84. Find all vectors that satisfy the equation
⟨
1
,
1
,
1
⟩
×
u
=
⟨
0
,
1
,
1
⟩
{\displaystyle \langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle }
85. Find the volume of the parallelepiped with edges given by position vectors
⟨
5
,
0
,
0
⟩
{\displaystyle \langle 5,0,0\rangle }
,
⟨
1
,
4
,
0
⟩
{\displaystyle \langle 1,4,0\rangle }
, and
⟨
2
,
2
,
7
⟩
{\displaystyle \langle 2,2,7\rangle }
140
{\displaystyle 140}
140
{\displaystyle 140}
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
F
=
⟨
1
,
2
,
3
⟩
{\displaystyle \mathbf {F} =\langle 1,2,3\rangle }
is applied to the wrench
n units away from the origin.
τ
=
⟨
0
,
−
3
n
,
2
n
⟩
{\displaystyle \mathbf {\tau } =\langle 0,-3n,2n\rangle }
±
⟨
0
,
−
3
,
2
⟩
{\displaystyle \pm \langle 0,-3,2\rangle }
τ
=
⟨
0
,
−
3
n
,
2
n
⟩
{\displaystyle \mathbf {\tau } =\langle 0,-3n,2n\rangle }
±
⟨
0
,
−
3
,
2
⟩
{\displaystyle \pm \langle 0,-3,2\rangle }
Prove the following identities or show them false by giving a counterexample.
89.
(
u
−
v
)
×
(
u
+
v
)
=
2
(
u
×
v
)
{\displaystyle (\mathbf {u} -\mathbf {v} )\times (\mathbf {u} +\mathbf {v} )=2(\mathbf {u} \times \mathbf {v} )}
Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}}
Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}}
Calculus of Vector-Valued Functions
edit
100. Differentiate
r
(
t
)
=
⟨
t
e
−
t
,
t
ln
t
,
t
cos
(
t
)
⟩
{\displaystyle \mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle }
.
⟨
e
−
t
−
t
e
−
t
,
ln
(
t
)
+
1
,
c
o
s
(
t
)
−
t
sin
(
t
)
⟩
{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle }
⟨
e
−
t
−
t
e
−
t
,
ln
(
t
)
+
1
,
c
o
s
(
t
)
−
t
sin
(
t
)
⟩
{\displaystyle \langle e^{-t}-te^{-t},\ln(t)+1,cos(t)-t\sin(t)\rangle }
101. Find a tangent vector for the curve
r
(
t
)
=
⟨
2
t
4
,
6
t
3
/
2
,
10
/
t
⟩
{\displaystyle \mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle }
at the point
t
=
1
{\displaystyle t=1}
.
⟨
8
,
9
,
−
10
⟩
{\displaystyle \langle 8,9,-10\rangle }
⟨
8
,
9
,
−
10
⟩
{\displaystyle \langle 8,9,-10\rangle }
102. Find the unit tangent vector for the curve
r
(
t
)
=
⟨
t
,
2
,
2
/
t
⟩
,
t
≠
0
{\displaystyle \mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0}
.
⟨
t
2
,
0
,
−
2
⟩
t
4
+
4
{\displaystyle \displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}
⟨
t
2
,
0
,
−
2
⟩
t
4
+
4
{\displaystyle \displaystyle {\frac {\langle t^{2},0,-2\rangle }{\sqrt {t^{4}+4}}}}
103. Find the unit tangent vector for the curve
r
(
t
)
=
⟨
sin
(
t
)
,
cos
(
t
)
,
e
−
t
⟩
,
t
∈
[
0
,
π
]
{\displaystyle \mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]}
at the point
t
=
0
{\displaystyle t=0}
.
⟨
1
,
0
,
−
1
⟩
2
{\displaystyle \displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}
⟨
1
,
0
,
−
1
⟩
2
{\displaystyle \displaystyle {\frac {\langle 1,0,-1\rangle }{\sqrt {2}}}}
104. Find
r
{\displaystyle \mathbf {r} }
if
r
′
(
t
)
=
⟨
t
,
cos
(
π
t
)
,
4
/
t
⟩
{\displaystyle \mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle }
and
r
(
1
)
=
⟨
2
,
3
,
4
⟩
{\displaystyle \mathbf {r} (1)=\langle 2,3,4\rangle }
.
⟨
2
t
3
/
2
+
4
3
,
sin
(
π
t
)
π
+
3
,
4
ln
|
t
|
+
4
⟩
{\displaystyle \displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle }
⟨
2
t
3
/
2
+
4
3
,
sin
(
π
t
)
π
+
3
,
4
ln
|
t
|
+
4
⟩
{\displaystyle \displaystyle \left\langle {\frac {2t^{3/2}+4}{3}},{\frac {\sin(\pi t)}{\pi }}+3,4\ln |t|+4\right\rangle }
120. Find velocity, speed, and acceleration of an object if the position is given by
r
(
t
)
=
⟨
3
sin
(
t
)
,
5
cos
(
t
)
,
4
sin
(
t
)
⟩
{\displaystyle \mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle }
.
v
=
⟨
3
cos
(
t
)
,
−
5
sin
(
t
)
,
4
cos
(
t
)
⟩
{\displaystyle \mathbf {v} =\langle 3\cos(t),-5\sin(t),4\cos(t)\rangle }
|
v
|
=
5
{\displaystyle |\mathbf {v} |=5}
a
=
⟨
−
3
sin
(
t
)
,
−
5
cos
(
t
)
,
−
4
sin
(
t
)
⟩
{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }
v
=
⟨
3
cos
(
t
)
,
−
5
sin
(
t
)
,
4
cos
(
t
)
⟩
{\displaystyle \mathbf {v} =\langle 3\cos(t),-5\sin(t),4\cos(t)\rangle }
|
v
|
=
5
{\displaystyle |\mathbf {v} |=5}
a
=
⟨
−
3
sin
(
t
)
,
−
5
cos
(
t
)
,
−
4
sin
(
t
)
⟩
{\displaystyle \mathbf {a} =\langle -3\sin(t),-5\cos(t),-4\sin(t)\rangle }
121. Find the velocity and the position vectors for
t
≥
0
{\displaystyle t\geq 0}
if the acceleration is given by
a
(
t
)
=
⟨
e
−
t
,
1
⟩
,
v
(
0
)
=
⟨
1
,
0
⟩
,
r
(
0
)
=
⟨
0
,
0
⟩
{\displaystyle \mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle }
.
v
(
t
)
=
⟨
2
−
e
−
t
,
t
⟩
{\displaystyle \mathbf {v} (t)=\langle 2-e^{-t},t\rangle }
r
(
t
)
=
⟨
e
−
t
+
2
t
−
1
,
t
2
/
2
⟩
{\displaystyle \mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle }
v
(
t
)
=
⟨
2
−
e
−
t
,
t
⟩
{\displaystyle \mathbf {v} (t)=\langle 2-e^{-t},t\rangle }
r
(
t
)
=
⟨
e
−
t
+
2
t
−
1
,
t
2
/
2
⟩
{\displaystyle \mathbf {r} (t)=\langle e^{-t}+2t-1,t^{2}/2\rangle }
Find the length of the following curves.
Parametrization and Normal Vectors
edit
142. Find a description of the curve that uses arc length as a parameter:
r
(
t
)
=
⟨
t
2
,
2
t
2
,
4
t
2
⟩
t
∈
[
1
,
4
]
.
{\displaystyle \mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].}
r
(
s
)
=
(
s
21
+
1
)
⟨
1
,
2
,
4
⟩
{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }
r
(
s
)
=
(
s
21
+
1
)
⟨
1
,
2
,
4
⟩
{\displaystyle \displaystyle \mathbf {r} (s)=\left({\frac {s}{\sqrt {21}}}+1\right)\langle 1,2,4\rangle }
143. Find the unit tangent vector
T and the principal unit normal vector
N for the curve
r
(
t
)
=
⟨
t
2
,
t
⟩
.
{\displaystyle \mathbf {r} (t)=\langle t^{2},t\rangle .}
Check that
T ⋅
N =0.
T
(
t
)
=
⟨
2
t
,
1
⟩
4
t
2
+
1
,
N
(
t
)
=
⟨
1
,
−
2
t
⟩
4
t
2
+
1
{\displaystyle \mathbf {T} (t)=\displaystyle {\frac {\langle 2t,1\rangle }{\sqrt {4t^{2}+1}}},\ \mathbf {N} (t)=\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}
T
(
t
)
=
⟨
2
t
,
1
⟩
4
t
2
+
1
,
N
(
t
)
=
⟨
1
,
−
2
t
⟩
4
t
2
+
1
{\displaystyle \mathbf {T} (t)=\displaystyle {\frac {\langle 2t,1\rangle }{\sqrt {4t^{2}+1}}},\ \mathbf {N} (t)=\displaystyle {\frac {\langle 1,-2t\rangle }{\sqrt {4t^{2}+1}}}}
Equations of Lines And Planes
edit
160. Find an equation of a plane passing through points
(
1
,
1
,
2
)
,
(
1
,
2
,
2
)
,
(
−
1
,
0
,
1
)
.
{\displaystyle (1,1,2),\ (1,2,2),\ (-1,0,1).}
x
−
2
z
+
3
=
0
{\displaystyle x-2z+3=0}
x
−
2
z
+
3
=
0
{\displaystyle x-2z+3=0}
161. Find an equation of a plane parallel to the plane 2x −y +z =1 passing through the point (0,2,-2)
2
x
−
y
+
z
+
4
=
0
{\displaystyle 2x-y+z+4=0}
2
x
−
y
+
z
+
4
=
0
{\displaystyle 2x-y+z+4=0}
162. Find an equation of the line perpendicular to the plane x +y +2z =4 passing through the point (5,5,5).
r
(
t
)
=
⟨
5
+
t
,
5
+
t
,
5
+
2
t
⟩
{\displaystyle \mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle }
r
(
t
)
=
⟨
5
+
t
,
5
+
t
,
5
+
2
t
⟩
{\displaystyle \mathbf {r} (t)=\langle 5+t,5+t,5+2t\rangle }
163. Find an equation of the line where planes x +2y −z =1 and x +y +z =1 intersect.
r
(
t
)
=
⟨
1
−
3
t
,
2
t
,
t
⟩
{\displaystyle \mathbf {r} (t)=\langle 1-3t,2t,t\rangle }
r
(
t
)
=
⟨
1
−
3
t
,
2
t
,
t
⟩
{\displaystyle \mathbf {r} (t)=\langle 1-3t,2t,t\rangle }
164. Find the angle between the planes x +2y −z =1 and x +y +z =1.
cos
−
1
2
18
{\displaystyle \cos ^{-1}{\frac {2}{\sqrt {18}}}}
cos
−
1
2
18
{\displaystyle \cos ^{-1}{\frac {2}{\sqrt {18}}}}
165. Find the distance from the point (3,4,5) to the plane x +y +z =1.
11
3
3
{\displaystyle {\frac {11}{3}}{\sqrt {3}}}
11
3
3
{\displaystyle {\frac {11}{3}}{\sqrt {3}}}
Limits And Continuity
edit
Evaluate the following limits.
180.
lim
(
x
,
y
)
→
(
1
,
−
2
)
y
2
+
2
x
y
y
+
2
x
{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}}
181.
lim
(
x
,
y
)
→
(
4
,
5
)
x
+
y
−
3
x
+
y
−
9
{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}}
At what points is the function f continuous?
183.
f
(
x
,
y
)
=
ln
(
x
2
+
y
2
)
x
−
y
+
1
{\displaystyle f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+1}}}
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.
lim
(
x
,
y
)
→
(
0
,
0
)
4
x
y
3
x
2
+
y
2
{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}}
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.
lim
(
x
,
y
)
→
(
0
,
0
)
x
3
−
y
2
x
3
+
y
2
{\displaystyle \displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}}
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
201. Find all three partial derivatives of the function
f
(
x
,
y
,
z
)
=
x
e
y
2
+
z
{\displaystyle \displaystyle f(x,y,z)=xe^{y^{2}+z}}
f
x
=
e
y
2
+
z
,
f
y
=
2
x
y
e
y
2
+
z
,
f
z
=
f
.
{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}
f
x
=
e
y
2
+
z
,
f
y
=
2
x
y
e
y
2
+
z
,
f
z
=
f
.
{\displaystyle \displaystyle f_{x}=e^{y^{2}+z},\ f_{y}=2xye^{y^{2}+z},\ f_{z}=f.}
Find the four second partial derivatives of the following functions.
202.
f
(
x
,
y
)
=
cos
(
x
y
)
{\displaystyle f(x,y)=\cos(xy)}
f
x
x
=
−
y
2
cos
(
x
y
)
,
f
y
y
=
−
x
2
cos
(
x
y
)
,
f
x
y
=
f
y
x
=
−
sin
(
x
y
)
−
x
y
cos
(
x
y
)
.
{\displaystyle f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).}
f
x
x
=
−
y
2
cos
(
x
y
)
,
f
y
y
=
−
x
2
cos
(
x
y
)
,
f
x
y
=
f
y
x
=
−
sin
(
x
y
)
−
x
y
cos
(
x
y
)
.
{\displaystyle f_{xx}=-y^{2}\cos(xy),\ f_{yy}=-x^{2}\cos(xy),\ f_{xy}=f_{yx}=-\sin(xy)-xy\cos(xy).}
203.
f
(
x
,
y
)
=
x
e
y
{\displaystyle f(x,y)=xe^{y}}
f
x
x
=
0
,
f
y
y
=
x
e
y
,
f
x
y
=
f
y
x
=
e
y
.
{\displaystyle f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.}
f
x
x
=
0
,
f
y
y
=
x
e
y
,
f
x
y
=
f
y
x
=
e
y
.
{\displaystyle f_{xx}=0,\ f_{yy}=xe^{y},\ f_{xy}=f_{yx}=e^{y}.}
Find
d
f
/
d
t
.
{\displaystyle df/dt.}
221.
f
(
x
,
y
)
=
x
2
+
y
2
,
x
(
t
)
=
cos
(
2
t
)
,
y
(
t
)
=
sin
(
2
t
)
{\displaystyle f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)}
222.
f
(
x
,
y
,
z
)
=
x
−
y
y
+
z
,
x
(
t
)
=
t
,
y
(
t
)
=
2
t
,
z
(
t
)
=
3
t
{\displaystyle \displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t}
Find
f
s
,
f
t
.
{\displaystyle f_{s},\ f_{t}.}
223.
f
(
x
,
y
)
=
sin
(
x
)
cos
(
2
y
)
,
x
=
s
+
t
,
y
=
s
−
t
{\displaystyle f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikibooks.org/v1/":): {\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)}
224.
f
(
x
,
y
,
z
)
=
x
−
z
y
+
z
,
x
(
t
)
=
s
+
t
,
y
(
t
)
=
s
t
,
z
(
t
)
=
s
−
t
{\displaystyle \displaystyle f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t}
Failed to parse (syntax error): {\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}
Failed to parse (syntax error): {\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}
225. The volume of a pyramid with a square base is
V
=
1
3
x
2
h
{\displaystyle V={\frac {1}{3}}x^{2}h}
, where
x is the side of the square base and
h is the height of the pyramid. Suppose that
x
(
t
)
=
t
t
+
1
{\displaystyle \displaystyle x(t)={\frac {t}{t+1}}}
and
h
(
t
)
=
1
t
+
1
{\displaystyle \displaystyle h(t)={\frac {1}{t+1}}}
for
t
≥
0.
{\displaystyle t\geq 0.}
Find
V
′
(
t
)
.
{\displaystyle V'(t).}
2
t
−
t
2
3
(
t
+
1
)
4
{\displaystyle \displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}}
2
t
−
t
2
3
(
t
+
1
)
4
{\displaystyle \displaystyle {\frac {2t-t^{2}}{3(t+1)^{4}}}}
Find an equation of a plane tangent to the given surface at the given point(s).
240.
x
y
sin
(
z
)
=
1
,
(
1
,
2
,
π
/
6
)
,
(
−
1
,
−
2
,
5
π
/
6
)
.
{\displaystyle xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).}
(
x
−
1
)
+
1
2
(
y
−
2
)
+
3
(
z
−
π
/
6
)
=
0
,
(
x
+
1
)
+
1
2
(
y
+
2
)
+
3
(
z
−
5
π
/
6
)
=
0
{\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}
(
x
−
1
)
+
1
2
(
y
−
2
)
+
3
(
z
−
π
/
6
)
=
0
,
(
x
+
1
)
+
1
2
(
y
+
2
)
+
3
(
z
−
5
π
/
6
)
=
0
{\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}
241.
z
=
x
2
e
x
−
y
,
(
2
,
2
,
4
)
,
(
−
1
,
−
1
,
1
)
.
{\displaystyle z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).}
−
8
(
x
−
2
)
+
4
(
y
−
2
)
+
z
−
4
=
0
,
x
+
y
+
z
+
1
=
0
{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}
−
8
(
x
−
2
)
+
4
(
y
−
2
)
+
z
−
4
=
0
,
x
+
y
+
z
+
1
=
0
{\displaystyle -8(x-2)+4(y-2)+z-4=0,\ x+y+z+1=0}
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.
260.
f
(
x
,
y
)
=
x
4
+
2
y
2
−
4
x
y
{\displaystyle f(x,y)=x^{4}+2y^{2}-4xy}
Local minima at (1,1) and (−1,−1), saddle at (0,0)
Local minima at (1,1) and (−1,−1), saddle at (0,0)
261.
f
(
x
,
y
)
=
tan
−
1
(
x
y
)
{\displaystyle f(x,y)=\tan ^{-1}(xy)}
262.
f
(
x
,
y
)
=
2
x
y
e
−
x
2
−
y
2
{\displaystyle f(x,y)=2xye^{-x^{2}-y^{2}}}
Saddle at (0,0), local maxima at
(
±
1
/
2
,
±
1
/
2
)
,
{\displaystyle (\pm 1/{\sqrt {2}},\pm 1/{\sqrt {2}}),}
(
±
1
/
2
,
∓
1
/
2
)
{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}
Saddle at (0,0), local maxima at
(
±
1
/
2
,
±
1
/
2
)
,
{\displaystyle (\pm 1/{\sqrt {2}},\pm 1/{\sqrt {2}}),}
(
±
1
/
2
,
∓
1
/
2
)
{\displaystyle (\pm 1/{\sqrt {2}},\mp 1/{\sqrt {2}})}
Find absolute maximum and minimum values of the function f on the set R .
263.
f
(
x
,
y
)
=
x
2
+
y
2
−
2
y
+
1
,
R
=
{
(
x
,
y
)
∣
x
2
+
y
2
≤
4
}
{\displaystyle f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}}
Maximum of 9 at (0,−2) and minimum of 0 at (0,1)
Maximum of 9 at (0,−2) and minimum of 0 at (0,1)
264.
f
(
x
,
y
)
=
x
2
+
y
2
−
2
x
−
2
y
,
{\displaystyle f(x,y)=x^{2}+y^{2}-2x-2y,}
R is a closed triangle with vertices (0,0), (2,0), and (0,2).
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)
Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)
265. Find the point on the plane x −y +z =2 closest to the point (1,1,1).
(
4
/
3
,
2
/
3
,
4
/
3
)
{\displaystyle (4/3,2/3,4/3)}
(
4
/
3
,
2
/
3
,
4
/
3
)
{\displaystyle (4/3,2/3,4/3)}
Double Integrals over Rectangular Regions
edit
Evaluate the given integral over the region R .
280.
∬
R
(
x
2
+
x
y
)
d
A
,
R
=
{
(
x
,
y
)
∣
x
∈
[
1
,
2
]
,
y
∈
[
−
1
,
1
]
}
{\displaystyle \displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}}
14
/
3
{\displaystyle 14/3}
14
/
3
{\displaystyle 14/3}
281.
∬
R
(
x
y
sin
(
x
2
)
)
d
A
,
R
=
{
(
x
,
y
)
∣
x
∈
[
0
,
π
/
2
]
,
y
∈
[
0
,
1
]
}
{\displaystyle \displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}}
1
/
4
{\displaystyle 1/4}
1
/
4
{\displaystyle 1/4}
282.
∬
R
x
(
1
+
x
y
)
2
d
A
,
R
=
{
(
x
,
y
)
∣
x
∈
[
0
,
4
]
,
y
∈
[
1
,
2
]
}
{\displaystyle \displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}}
ln
(
5
/
3
)
{\displaystyle \ln(5/3)}
ln
(
5
/
3
)
{\displaystyle \ln(5/3)}
Evaluate the given iterated integrals.
Double Integrals over General Regions
edit
Evaluate the following integrals.
300.
∬
R
x
y
d
A
,
{\displaystyle \displaystyle \iint _{R}xydA,}
R is bounded by
x =0,
y =2
x +1, and
y =5−2
x .
2
{\displaystyle 2}
2
{\displaystyle 2}
301.
∬
R
(
x
+
y
)
d
A
,
{\displaystyle \displaystyle \iint _{R}(x+y)dA,}
R is in the first quadrant and bounded by
x =0,
y
=
x
2
,
{\displaystyle y=x^{2},}
and
y
=
8
−
x
2
.
{\displaystyle y=8-x^{2}.}
152
/
3
{\displaystyle 152/3}
152
/
3
{\displaystyle 152/3}
Use double integrals to compute the volume of the given region.
Double Integrals in Polar Coordinates
edit
323. Evaluate
∬
R
x
−
y
x
2
+
y
2
+
1
d
A
{\displaystyle \displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA}
if
R is the unit disk centered at the origin.
0
{\displaystyle 0}
0
{\displaystyle 0}
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.
8
{\displaystyle 8}
8
{\displaystyle 8}
342. Find the volume of the solid in the first octant bounded by the cylinder
z
=
sin
(
y
)
{\displaystyle z=\sin(y)}
for
y
∈
[
0
,
π
]
{\displaystyle y\in [0,\pi ]}
, and the planes
y =
x and
x =0.
π
{\displaystyle \pi }
π
{\displaystyle \pi }
344. Rewrite the integral
∫
0
1
∫
−
2
2
∫
0
4
−
y
2
d
z
d
y
d
x
{\displaystyle \displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx}
in the order
dydzdx .
∫
0
1
∫
0
2
∫
−
4
−
z
2
4
−
z
2
d
y
d
z
d
x
{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}
∫
0
1
∫
0
2
∫
−
4
−
z
2
4
−
z
2
d
y
d
z
d
x
{\displaystyle \displaystyle \int _{0}^{1}\int _{0}^{2}\int _{-{\sqrt {4-z^{2}}}}^{\sqrt {4-z^{2}}}dydzdx}
Cylindrical And Spherical Coordinates
edit
361. Find the mass of the solid cylinder
D
=
{
(
r
,
θ
,
z
)
∣
r
∈
[
0
,
3
]
,
z
∈
[
0
,
2
]
}
{\displaystyle D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}}
given the density function
δ
(
r
,
θ
,
z
)
=
5
e
−
r
2
{\displaystyle \delta (r,\theta ,z)=5e^{-r^{2}}}
10
π
(
1
−
e
−
9
)
{\displaystyle 10\pi (1-e^{-9})}
10
π
(
1
−
e
−
9
)
{\displaystyle 10\pi (1-e^{-9})}
362. Use a triple integral to find the volume of the region bounded by the plane
z =0 and the hyperboloid
z
=
17
−
1
+
x
2
+
y
2
{\displaystyle z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}}
2
π
(
1
+
7
17
)
3
{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}
2
π
(
1
+
7
17
)
3
{\displaystyle \displaystyle {\frac {2\pi (1+7{\sqrt {17}})}{3}}}
363. If
D is a unit ball, use a triple integral in spherical coordinates to evaluate
∭
D
(
x
2
+
y
2
+
z
2
)
5
/
2
d
V
{\displaystyle \iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV}
π
/
2
{\displaystyle \pi /2}
π
/
2
{\displaystyle \pi /2}
364. Find the mass of a solid cone
{
(
ρ
,
ϕ
,
θ
)
∣
ϕ
≤
π
/
3
,
z
∈
[
0
,
4
]
}
{\displaystyle \{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}}
if the density function is
δ
(
ρ
,
ϕ
,
θ
)
=
5
−
z
{\displaystyle \delta (\rho ,\phi ,\theta )=5-z}
128
π
{\displaystyle 128\pi }
128
π
{\displaystyle 128\pi }
Center of Mass and Centroid
edit
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.
⟨
3
,
5
,
7
⟩
4
{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}
⟨
3
,
5
,
7
⟩
4
{\displaystyle {\frac {\langle 3,5,7\rangle }{4}}}
384. Find the centroid of the region in the first quadrant bounded by
y
=
ln
(
x
)
{\displaystyle y=\ln(x)}
,
y
=
0
{\displaystyle y=0}
, and
x
=
e
{\displaystyle x=e}
.
(
(
e
2
+
1
)
/
4
,
e
/
2
−
1
)
{\displaystyle ((e^{2}+1)/4,e/2-1)}
(
(
e
2
+
1
)
/
4
,
e
/
2
−
1
)
{\displaystyle ((e^{2}+1)/4,e/2-1)}
385. Find the center of mass for the region
{
(
x
,
y
)
∣
x
∈
[
0
,
4
]
,
y
∈
[
0
,
2
]
}
{\displaystyle \{(x,y)\mid x\in [0,4],y\in [0,2]\}}
, with the density
ρ
(
x
,
y
)
=
1
+
x
/
2.
{\displaystyle \rho (x,y)=1+x/2.}
(
7
/
3
,
1
)
{\displaystyle (7/3,1)}
(
7
/
3
,
1
)
{\displaystyle (7/3,1)}
386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density
ρ
(
x
,
y
)
=
1
+
x
+
y
.
{\displaystyle \rho (x,y)=1+x+y.}
(
16
/
11
,
16
/
11
)
{\displaystyle (16/11,16/11)}
(
16
/
11
,
16
/
11
)
{\displaystyle (16/11,16/11)}
One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.
402. Find and sketch the gradient field
F
=
∇
ϕ
{\displaystyle \mathbf {F} =\nabla \phi }
for the potential function
ϕ
(
x
,
y
)
=
sin
(
x
)
sin
(
y
)
{\displaystyle \phi (x,y)=\sin(x)\sin(y)}
for
|
x
|
≤
π
{\displaystyle |x|\leq \pi }
and
|
y
|
≤
π
{\displaystyle |y|\leq \pi }
.
∇
ϕ
(
x
,
y
)
=
⟨
cos
(
x
)
sin
(
y
)
,
sin
(
x
)
cos
(
y
)
⟩
{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }
∇
ϕ
(
x
,
y
)
=
⟨
cos
(
x
)
sin
(
y
)
,
sin
(
x
)
cos
(
y
)
⟩
{\displaystyle \nabla \phi (x,y)=\langle \cos(x)\sin(y),\sin(x)\cos(y)\rangle }
403. Find the gradient field
F
=
∇
ϕ
{\displaystyle \mathbf {F} =\nabla \phi }
for the potential function
ϕ
(
x
,
y
,
z
)
=
e
−
z
sin
(
x
+
y
)
{\displaystyle \phi (x,y,z)=e^{-z}\sin(x+y)}
F
=
e
−
z
⟨
cos
(
x
+
y
)
,
cos
(
x
+
y
)
,
−
sin
(
x
+
y
)
⟩
{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }
F
=
e
−
z
⟨
cos
(
x
+
y
)
,
cos
(
x
+
y
)
,
−
sin
(
x
+
y
)
⟩
{\displaystyle \mathbf {F} =e^{-z}\left\langle \cos(x+y),\cos(x+y),-\sin(x+y)\right\rangle }
420. Evaluate
∫
C
(
x
2
+
y
2
)
d
s
{\displaystyle \int _{C}(x^{2}+y^{2})ds}
if
C is the line segment from (0,0) to (5,5)
250
2
3
{\displaystyle {\frac {250{\sqrt {2}}}{3}}}
250
2
3
{\displaystyle {\frac {250{\sqrt {2}}}{3}}}
421. Evaluate
∫
C
(
x
2
+
y
2
)
d
s
{\displaystyle \int _{C}(x^{2}+y^{2})ds}
if
C is the circle of radius 4 centered at the origin
128
π
{\displaystyle 128\pi }
128
π
{\displaystyle 128\pi }
422. Evaluate
∫
C
(
y
−
z
)
d
s
{\displaystyle \int _{C}(y-z)ds}
if
C is the helix
r
(
t
)
=
⟨
3
cos
(
t
)
,
3
sin
(
t
)
,
t
⟩
,
t
∈
[
0
,
2
π
]
{\displaystyle \mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]}
−
2
10
π
2
{\displaystyle -2{\sqrt {10}}\pi ^{2}}
−
2
10
π
2
{\displaystyle -2{\sqrt {10}}\pi ^{2}}
423. Evaluate
∫
C
F
⋅
d
r
{\displaystyle \int _{C}\mathbf {F} \cdot d\mathbf {r} }
if
F
=
⟨
x
,
y
⟩
{\displaystyle \mathbf {F} =\langle x,y\rangle }
and
C is the arc of the parabola
r
(
t
)
=
⟨
4
t
,
t
2
⟩
,
t
∈
[
0
,
1
]
{\displaystyle \mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]}
17
/
2
{\displaystyle 17/2}
17
/
2
{\displaystyle 17/2}
Conservative Vector Fields
edit
Determine if the following vector fields are conservative on
R
2
.
{\displaystyle \mathbb {R} ^{2}.}
440.
⟨
−
y
,
x
+
y
⟩
{\displaystyle \langle -y,x+y\rangle }
441.
⟨
2
x
3
+
x
y
2
,
2
y
3
+
x
2
y
⟩
{\displaystyle \langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle }
Determine if the following vector fields are conservative on their respective domains in
R
3
.
{\displaystyle \mathbb {R} ^{3}.}
442.
⟨
y
,
x
,
1
⟩
{\displaystyle \langle y,x,1\rangle }
ϕ
(
x
,
y
,
z
)
=
x
y
+
z
{\displaystyle \phi (x,y,z)=xy+z}
ϕ
(
x
,
y
,
z
)
=
x
y
+
z
{\displaystyle \phi (x,y,z)=xy+z}
443.
⟨
x
3
,
2
y
,
−
z
3
⟩
{\displaystyle \langle x^{3},2y,-z^{3}\rangle }
ϕ
(
x
,
y
,
z
)
=
(
x
4
+
4
y
2
−
z
4
)
/
4
{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}
ϕ
(
x
,
y
,
z
)
=
(
x
4
+
4
y
2
−
z
4
)
/
4
{\displaystyle \phi (x,y,z)=(x^{4}+4y^{2}-z^{4})/4}
460. Evaluate the circulation of the field
F
=
⟨
2
x
y
,
x
2
−
y
2
⟩
{\displaystyle \mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle }
over the boundary of the region above
y =0 and below
y =
x (2-
x ) in two different ways, and compare the answers.
0
{\displaystyle 0}
0
{\displaystyle 0}
461. Evaluate the circulation of the field
F
=
⟨
0
,
x
2
+
y
2
⟩
{\displaystyle \mathbf {F} =\langle 0,x^{2}+y^{2}\rangle }
over the unit circle centered at the origin in two different ways, and compare the answers.
0
{\displaystyle 0}
0
{\displaystyle 0}
462. Evaluate the flux of the field
F
=
⟨
y
,
−
x
⟩
{\displaystyle \mathbf {F} =\langle y,-x\rangle }
over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.
0
{\displaystyle 0}
0
{\displaystyle 0}
482. Find the curl of
⟨
x
2
−
y
2
,
x
y
,
z
⟩
{\displaystyle \langle x^{2}-y^{2},xy,z\rangle }
⟨
0
,
0
,
3
y
⟩
{\displaystyle \langle 0,0,3y\rangle }
⟨
0
,
0
,
3
y
⟩
{\displaystyle \langle 0,0,3y\rangle }
484. Prove that the general rotation field
F
=
a
×
r
{\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} }
, where
a
{\displaystyle \mathbf {a} }
is a non-zero constant vector and
r
=
⟨
x
,
y
,
z
⟩
{\displaystyle \mathbf {r} =\langle x,y,z\rangle }
, has zero divergence, and the curl of
F
{\displaystyle \mathbf {F} }
is
2
a
{\displaystyle 2\mathbf {a} }
.
If
a
=
⟨
a
1
,
a
2
,
a
3
⟩
{\displaystyle \mathbf {a} =\langle a_{1},a_{2},a_{3}\rangle }
F
=
a
×
r
=
⟨
a
2
z
−
a
3
y
,
a
3
x
−
a
1
z
,
a
1
y
−
a
2
x
⟩
=
⟨
f
,
g
,
h
⟩
{\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle }
∇
⋅
F
=
f
x
+
g
y
+
h
z
=
0
+
0
+
0
=
0
,
{\displaystyle \nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,}
∇
×
F
=
⟨
h
y
−
g
z
,
f
z
−
h
x
,
g
x
−
f
y
⟩
=
⟨
a
1
+
a
1
,
a
2
+
a
2
,
a
3
+
a
3
⟩
=
2
a
.
{\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} .}
If
a
=
⟨
a
1
,
a
2
,
a
3
⟩
{\displaystyle \mathbf {a} =\langle a_{1},a_{2},a_{3}\rangle }
F
=
a
×
r
=
⟨
a
2
z
−
a
3
y
,
a
3
x
−
a
1
z
,
a
1
y
−
a
2
x
⟩
=
⟨
f
,
g
,
h
⟩
{\displaystyle \mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle }
∇
⋅
F
=
f
x
+
g
y
+
h
z
=
0
+
0
+
0
=
0
,
{\displaystyle \nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,}
∇
×
F
=
⟨
h
y
−
g
z
,
f
z
−
h
x
,
g
x
−
f
y
⟩
=
⟨
a
1
+
a
1
,
a
2
+
a
2
,
a
3
+
a
3
⟩
=
2
a
.
{\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} .}
500. Give a parametric description of the plane
2
x
−
4
y
+
3
z
=
16.
{\displaystyle 2x-4y+3z=16.}
⟨
u
,
v
,
(
16
−
2
u
+
4
v
)
/
3
⟩
,
u
,
v
∈
R
{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }
⟨
u
,
v
,
(
16
−
2
u
+
4
v
)
/
3
⟩
,
u
,
v
∈
R
{\displaystyle \langle u,v,(16-2u+4v)/3\rangle ,\ u,v\in \mathbb {R} }
501. Give a parametric description of the hyperboloid
z
2
=
1
+
x
2
+
y
2
.
{\displaystyle z^{2}=1+x^{2}+y^{2}.}
⟨
v
2
−
1
cos
(
u
)
,
v
2
−
1
sin
(
u
)
,
v
⟩
,
u
∈
[
0
,
2
π
]
,
|
v
|
≥
1
{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}
⟨
v
2
−
1
cos
(
u
)
,
v
2
−
1
sin
(
u
)
,
v
⟩
,
u
∈
[
0
,
2
π
]
,
|
v
|
≥
1
{\displaystyle \langle {\sqrt {v^{2}-1}}\cos(u),{\sqrt {v^{2}-1}}\sin(u),v\rangle ,\ u\in [0,2\pi ],\ |v|\geq 1}
502. Integrate
f
(
x
,
y
,
z
)
=
x
y
{\displaystyle f(x,y,z)=xy}
over the portion of the plane
z =2−
x −
y in the first octant.
2
/
3
{\displaystyle 2/{\sqrt {3}}}
2
/
3
{\displaystyle 2/{\sqrt {3}}}
504. Find the flux of the field
F
=
⟨
x
,
y
,
z
⟩
{\displaystyle \mathbf {F} =\langle x,y,z\rangle }
across the surface of the cone
z
2
=
x
2
+
y
2
,
z
∈
[
0
,
1
]
,
{\displaystyle z^{2}=x^{2}+y^{2},\ z\in [0,1],}
with normal vectors pointing in the positive
z direction.
0
{\displaystyle 0}
0
{\displaystyle 0}
505. Find the flux of the field
F
=
⟨
−
y
,
z
,
1
⟩
{\displaystyle \mathbf {F} =\langle -y,z,1\rangle }
across the surface
y
=
x
2
,
z
∈
[
0
,
4
]
,
x
∈
[
0
,
1
]
,
{\displaystyle y=x^{2},\ z\in [0,4],\ x\in [0,1],}
with normal vectors pointing in the positive
y direction.
−
10
{\displaystyle -10}
−
10
{\displaystyle -10}
520. Use a surface integral to evaluate the circulation of the field
F
=
⟨
x
2
−
z
2
,
y
,
2
x
z
⟩
{\displaystyle \mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle }
on the boundary of the plane
z
=
4
−
x
−
y
{\displaystyle z=4-x-y}
in the first octant.
−
128
3
{\displaystyle {\frac {-128}{3}}}
−
128
3
{\displaystyle {\frac {-128}{3}}}
522. Use a line integral to find
∬
S
(
∇
×
F
)
⋅
n
d
S
{\displaystyle \iint _{S}(\nabla \times F)\cdot \mathbf {n} dS}
where
F
=
⟨
x
,
y
,
z
⟩
{\displaystyle \mathbf {F} =\langle x,y,z\rangle }
,
S
{\displaystyle S}
is the upper half of the ellipsoid
x
2
4
+
y
2
9
+
z
2
=
1
{\displaystyle {\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1}
, and
n
{\displaystyle \mathbf {n} }
points in the direction of the
z -axis.
0
{\displaystyle 0}
0
{\displaystyle 0}
523. Use a line integral to find
∬
S
(
∇
×
F
)
⋅
n
d
S
{\displaystyle \iint _{S}(\nabla \times F)\cdot \mathbf {n} dS}
where
F
=
⟨
2
y
,
−
z
,
x
−
y
−
z
⟩
{\displaystyle \mathbf {F} =\langle 2y,-z,x-y-z\rangle }
,
S
{\displaystyle S}
is the part of the sphere
x
2
+
y
2
+
z
2
=
25
{\displaystyle x^{2}+y^{2}+z^{2}=25}
for
3
≤
z
≤
5
{\displaystyle 3\leq z\leq 5}
, and
n
{\displaystyle \mathbf {n} }
points in the direction of the
z -axis.
−
32
π
{\displaystyle -32\pi }
−
32
π
{\displaystyle -32\pi }
Compute the net outward flux of the given field across the given surface.
540.
F
=
⟨
x
,
−
2
y
,
3
z
⟩
{\displaystyle \mathbf {F} =\langle x,-2y,3z\rangle }
,
S
{\displaystyle S}
is a sphere of radius
6
{\displaystyle {\sqrt {6}}}
centered at the origin.
16
6
π
{\displaystyle 16{\sqrt {6}}\pi }
16
6
π
{\displaystyle 16{\sqrt {6}}\pi }
542.
F
=
⟨
y
+
z
,
x
+
z
,
x
+
y
⟩
{\displaystyle \mathbf {F} =\langle y+z,x+z,x+y\rangle }
,
S
{\displaystyle S}
is the boundary of the cube
{
(
x
,
y
,
z
)
∣
|
x
|
≤
1
,
|
y
|
≤
1
,
|
z
|
≤
1
}
{\displaystyle \{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}}
0
{\displaystyle 0}
0
{\displaystyle 0}
543.
F
=
⟨
x
,
y
,
z
⟩
{\displaystyle \mathbf {F} =\langle x,y,z\rangle }
,
S
{\displaystyle S}
is the surface of the region bounded by the paraboloid
z
=
4
−
x
2
−
y
2
{\displaystyle z=4-x^{2}-y^{2}}
and the
xy -plane.
24
π
{\displaystyle 24\pi }
24
π
{\displaystyle 24\pi }
544.
F
=
⟨
z
−
x
,
x
−
y
,
2
y
−
z
⟩
{\displaystyle \mathbf {F} =\langle z-x,x-y,2y-z\rangle }
,
S
{\displaystyle S}
is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.
−
224
π
{\displaystyle -224\pi }
−
224
π
{\displaystyle -224\pi }
545.
F
=
⟨
x
,
2
y
,
3
z
⟩
{\displaystyle \mathbf {F} =\langle x,2y,3z\rangle }
,
S
{\displaystyle S}
is the boundary of the region between the cylinders
x
2
+
y
2
=
1
{\displaystyle x^{2}+y^{2}=1}
and
x
2
+
y
2
=
4
{\displaystyle x^{2}+y^{2}=4}
and cut off by planes
z
=
0
{\displaystyle z=0}
and
z
=
8
{\displaystyle z=8}
144
π
{\displaystyle 144\pi }
144
π
{\displaystyle 144\pi }
![{\displaystyle \left\langle {\frac {4}{5}}c,-{\frac {3}{5}}c,{\sqrt {1-c^{2}}}\right\rangle ,\ c\in [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec58739fb84b755913f1b1240cecc19cacfcaa51)
![{\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)
![{\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)
![{\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)
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