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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
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)} Polar Coordinates
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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
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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
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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 (unknown function "\begin{eqnarray}"): {\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 (unknown function "\begin{eqnarray}"): {\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]} Cross Product
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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
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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 } Motion in Space
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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 } Length of Curves
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Find the length of the following curves.
Parametrization and Normal Vectors
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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
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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
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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
Partial Derivatives
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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}.} Chain Rule
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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 (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 \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}
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 \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}}}} Tangent Planes
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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
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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
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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
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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
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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} Triple Integrals
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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
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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
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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.
⟨ 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)} Vector Fields
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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 } Line Integrals
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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
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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} Green's Theorem
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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} Divergence And Curl
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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} .} Surface Integrals
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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} Stokes' Theorem
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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 } Divergence Theorem
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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)
![{\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)
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