Exam 13: Vector Calculus

Find the work done by the force field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ on a particle that moves along the circle C: $x = \cos t , y = \sin t , 0 \leq t \leq 2 \pi$ .

Let $\mathbf { F } ( x , y , z ) = 2 x y ^ { 3 } z ^ { 4 } \mathbf { i } + 3 x ^ { 2 } y ^ { 2 } z ^ { 4 } \mathbf { j } + 4 x ^ { 2 } y ^ { 3 } z ^ { 3 } \mathbf { k }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , \quad 0 \leq t \leq 2 \pi$ .

Let $\mathbf { F } ( x , y , z ) = y ^ { 2 } \mathbf { i } + x \mathbf { j }$ . Find the value of the curl of F at the point (1,1,3).

Determine whether $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ is conservative and if so, find a potential function.

For what value of the constant $b$ is the vector field $\mathbf { F } = b x y ^ { 2 } \mathbf { i } + x ^ { 2 } y \mathbf { j }$ conservative?

Find a formula for the vector field graphed below. (There are many possible answers.)

Evaluate the line integral $\int _ { C } x y d x + y z d y + z x d x$ : (a) if $C$ is the line segment from $( 1,1,1 )$ to $( 3,3,3 )$ .(b) if $C$ is the line segment from $( 3,3,3 )$ to $( 1,1,1 )$ .

Use the Divergence Theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = x ( y - 1 ) \mathbf { i } + 2 y z \mathbf { j } - \left( z ^ { 2 } + y z \right) \mathbf { k }$ and S is the surface of the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ , bounded by the planes $z = 0$ and $z = 3$ .

Let $\mathbf { F } ( x , y , z ) = z \mathbf { i } + x \mathbf { j } + y \mathbf { k }$ . Find the divergence of F.

Let $\phi ( x , y ) = 3 x ^ { 2 } y ^ { 3 }$ . Note that $\nabla \phi = \left\langle 6 x y ^ { 3 } , 9 x ^ { 2 } y ^ { 2 } \right\rangle$ . Evaluate the line integral $\int _ { C } 6 x y ^ { 3 } d x + 9 x ^ { 2 } y ^ { 2 } d y$ where C is the curve $x = t e ^ { t ^ { 3 } - 1 } , \mathrm { y } = ( 2 t + 1 ) \cos ( 2 \pi t )$ , $0 \leq t \leq 1$ .

Find a function f such that $\mathbf { F } = \nabla f$ and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the curve C. $\mathbf { F } ( x , y ) = y \mathbf { i } + x \mathbf { j };$ C is the arc of the curve $y = x ^ { 4 } - x ^ { 3 }$ from $( 1,0 )$ to $( 2,8 )$ .

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $\mathbf { F } ( x , y ) = e ^ { x } \mathbf { i } + x y \mathbf { j }$ and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j } , \quad 0 \leq t \leq 1$ .

Evaluate $\oint _ { C } \left( x \cos x + e ^ { \sin x } + 3 y \right) d x + \left[ 5 x + \left( 1 + y ^ { 2 } \right) ^ { 3 } + \cos \left( e ^ { y } + y ^ { 2 } \right) \right] d y$ where C is the positively-oriented boundary of the half disk D: $x ^ { 2 } + y ^ { 2 } \leq 2 , y \geq 0$ .

Find the work done by the force F $= ( 2 x + y ) \mathbf { i } + ( x y ) \mathbf { j }$ in moving an object from $( 1,0 )$ to $( 2,3 )$ along the path $C$ given by $x = t + 1 , y = 3 t$ ..

Evaluate the line integral $\int _ { C } e ^ { x ^ { 2 } y ^ { 2 } z ^ { 2 } } d x + \sin ( x y z ) d y + \ln ( 1 + x y ) d z$ where $C$ is the line segment from $( 0,0,0 )$ to $( 0,0,1 )$

Use Green's Theorem to find the area of the region formed by the intersection of $x ^ { 2 } + y ^ { 2 } \leq 2$ and $y \geq 1$ .

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ is the curve $C$ is given by $C : x = 5 \cos t , y = 5 \sin t , 0 \leq t \leq 2 \pi$ .

Let $f ( x , y , z ) = \ln \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right)$ compute $\nabla ^ { 2 } f$ .

Evaluate the line integral $\int _ { c } x d s$ , where $C$ is the curve $x = t , \quad y = t , \quad 0 \leq t \leq 1$ .

Let $\mathbf { F } ( x , y , z ) = x \mathbf { j }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the elliptical path $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + \cos t \mathbf { k } , 0 \leq \mathrm { t } \leq 2 \pi$ .