Exam 13: Vector Calculus

A wire in the shape of the curve $x = t ^ { 3 } , y = 2 t ^ { \frac { 3 } { 2 } } , z = 3 t + 1,0 \leq t \leq 1$ has density function $p ( x , y , z ) = y ^ { 2 } + 1$ . Find the mass of the wire.

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

Use Green's Theorem to compute the area of the region enclosed by the curve $x = \cos ^ { 3 } t , y = \sin ^ { 3 } t , 0 \leq t \leq 2 \pi$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { i } + \frac { y } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { j } + \frac { z } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { k }$ and S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ .

Sketch the vector field $\mathbf { F } ( x , y ) = - x \mathbf { i } + y \mathbf { j }$ .

Find a function $f ( x , y )$ such that $\nabla f = \left( x ^ { 2 } + y \right) \mathbf { i } + ( x + y ) \mathbf { j }$ .

Evaluate $\int _ { c } x y ^ { 2 } d s$ where the $C$ is given by $C : x = 2 \cos t , \quad y = 2 \sin t , z = \sqrt { 5 t } , 0 \leq t \leq \frac { \pi } { 2 }$ .

Determine the points $( x , y , z )$ where the gradient field $\nabla f ( x , y , z )$ for $f ( x , y , z ) = x y + x z + y z$ has $Z$ -component $0$ .

Use Stokes' Theorem to evaluate $\int _ { C } x y d x + y z d y + z x d z$ , where C is the triangle with vertices $( 1,0,0 ) , ( 0,1,0 )$ , and $( 0,0,1 )$ , oriented counterclockwise as viewed from above.

Evaluate the line integral $\int _ { C } x y d x + ( x + y ) d y$ where $C$ is the curve $x = t ^ { 2 } , y = t ^ { 3 } , 0 \leq t \leq 1$

Evaluate the line integral $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = x y ^ { 2 } \mathbf { i } + y \mathbf { j }$ and the curve $C$ is the shortest path from $( 0,0 )$ to $( 0,1 )$ to $( 1.1 )$ .

Use Stokes' Theorem to evaluate $\int _ { C } ( x z + 2 y ) d x + \left( x ^ { 2 } + 5 y z \right) d y + \left( y ^ { 2 } + z \right) d z$ , where C is the curve of intersection of the paraboloid $z = x ^ { 2 } + 5 y ^ { 2 }$ and the cylinder $( x - 4 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 16$ , oriented counterclockwise as viewed from above.

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

Evaluate the line integral $\int _ { C } x y d x + z d y + y d z$ , where the curve C is given in the figure below.

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. $\mathbf { F } ( x , y , z ) = 2 x y ^ { 3 } z ^ { 4 } \mathbf { i } + 3 x ^ { 2 } y ^ { 2 } z ^ { 4 } \mathbf { j } + 4 x ^ { 2 } y ^ { 3 } z ^ { 3 } \mathbf { k }$ ; $\text { C. } x = t$ , $y = t ^ { 2 }$ , $z = t ^ { 3 }$ , $0 \leq t \leq 2$

Sketch the vector field F where F $( x , y , z ) = z \mathbf { k }$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is the boundary surface of the region outside the sphere $\mathrm { x } ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ and inside the ball $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4$ and $\mathbf { F } = e ^ { y ^ { 2 } } \mathbf { i } + \left( 5 y + e ^ { x ^ { 2 } } \right) \mathbf { j } + ( - 3 z + 2 x ) \mathbf { k }$ .

Find the work done by the force field $\mathbf { F } ( x , y ) = x y \mathbf { i } + x ^ { 2 } \mathbf { j }$ on a particle that moves along the curve $x = y ^ { 2 }$ from $( 0,0 )$ to $( 1.1 )$ .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } x ^ { 2 } y d x + x y ^ { 5 } d y$ , where C is the square with vertices $( \pm 1 , \pm 1 )$ .

Determine whether or not each of the following vector fields is conservative. Justify your answer.(a) (b) (c)