Exam 17: Integrals and Vector Fields

Solve the problem. -Assuming $C$ is a simple closed path, what is special about the integral $\left. \left. \int _ { C } \left( 6 x + 8 e ^ { 9 x } \cos 9 y \right) \right) d x + \left( 3 x + 8 e ^ { 9 x } \sin 9 y \right) \right) d y ?$ Give reasons for your answer.

$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = ( 3 + 3 t ) i + j - 3 t k , 0 \leq t \leq 1 ; \delta ( x , y , z ) = x + z ^ { 2 }$

Find the surface area of the surface S. - $\mathrm { S }$ is the surface $\frac { 7 } { 2 } \mathrm { x } ^ { 2 } + 7 \mathrm { z } = 0$ that lies above the region bounded by the $x$ -axis, $x = \sqrt { 15 }$ , and $y = x$ .

Solve the problem. -Imagine a force field in which the force is always parallel to dr. What is special about the work done in moving a particle in such a field?

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j }$ ; the curve $C$ is the closed counterclockwise path around the triangle with vertices at $( 0,0 )$ , $( 3,0 )$ , and $( 0,1 )$

Calculate the area of the surface S. - $S$ is the portion of the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ that lies between $z = 3$ and $z = 4$ .

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = x ^ { 2 } + y ^ { 2 } , C : y = - 4 x - 3,0 \leq x \leq 3$

Find the gradient field of the function. - $f ( x , y , z ) = \frac { x z + x y + y z } { x y z }$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } ( x , y , z ) = z k , S$ is the surface of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ in the first octant, direction away from the origin

Find the gradient field of the function. - $f ( x , y , z ) = \ln \left( \frac { ( x + y ) ^ { 3 } } { z ^ { 10 } } \right) + z ^ { 9 }$

Find the potential function f for the field F. - $\mathbf { F } = \sec ^ { 2 } x \sin ^ { 2 } y \mathbf { i } + 2 \tan x \sin y \cos y \mathbf { j } + 8 z ^ { 7 } \mathbf { k }$

$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = 4 i + ( 3 - 4 t ) j + 3 t k , 0 \leq t \leq 2 \pi ; \delta = 4 ( 1 + \sin 10 t )$

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = y \mathbf { i } - \mathrm { xj } ; \text { the curve } \mathrm { C } \text { is the circle } ( x + 6 ) ^ { 2 } + ( \mathrm { y } + 9 ) ^ { 2 } = 144$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = \mathrm { x } ^ { 2 } \mathrm { yi } - 3 x \mathrm { y } ^ { 2 } \mathbf { j } + \ln \mathrm { zk } ; \mathrm { S } : \mathrm { r } ( \mathrm { r } , \theta ) = \mathrm { r } \cos \theta \mathbf { i } + \mathrm { r } \sin \theta \mathbf { j } + 2 \mathrm { rk } , 0 \leq \mathrm { r } \leq 2$ and $0 \leq \theta \leq 2 \pi$

Find the potential function f for the field F. - $F = 2 x e ^ { x ^ { 2 } + y ^ { 2 } } i + 2 y e ^ { x ^ { 2 } + y ^ { 2 } } j$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = - 12 \mathrm { xz }^ 5 \mathrm { i } + 2 \mathrm { yj } + 2 \mathrm { z } ^ { 6 } \mathbf { k } ;$ D: the solid wedge cut from the first quadrant by the plane $\mathrm { z } = 4 \mathrm { y }$ and the elliptic cylinder $x ^ { 2 } + 25 y ^ { 2 } = 49$

Find the limit. - $r ( t ) = t j ; - 2 \leq t \leq 2$

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = \ln \left( x ^ { 2 } + y ^ { 2 } \right) i + \tan ^ { - 1 } \left( \frac { x } { y } \right) ; \quad C \text { is the region defined by the polar coordinate inequalities } 3 \leq r \leq 6 \text { and } 0 \leq \theta \leq \pi$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = 5 \mathrm { y } \mathbf { i } + 2 \mathrm { x } \mathbf { j } + \cos ( \mathrm { z } ) \mathbf { k } ; \mathrm { S } : \mathbf { r } ( \mathrm { r } , \theta ) = 2 \sin \varphi \cos \theta \mathbf { i } + 2 \sin \varphi \sin \theta \mathbf { j } + 2 \cos \varphi \mathbf { k } , 0 \leq \theta \leq 2 \pi$ and $0 \leq \varphi \leq \frac { \pi } { 2 }$

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = x , C : y = x ^ { 2 } , 0 \leq x \leq \frac { \sqrt { 35 } } { 2 }$