Exam 13: Vector Calculus

Use Stokes' Theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = z ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + x y \mathbf { k }$ and C is the triangle with vertices $( 1,0,0 ) , ( 0,1,0 )$ , and $( 0,0,2 )$

Find the work done by the force field $\mathbf { F } ( x , y ) = \nabla \left( x ^ { 2 } + y ^ { 2 } \right) ^ { - 1 / 2 }$ on a particle that moves from the point $( 3,4 )$ to the point $( 0,2 )$ .

Find the gradient vector field of $f ( x , y , z ) = x y z$ .

Let $\mathbf { F } ( x , y , z ) = ( z - 4 y \sin x ) \mathbf { i } + x z \mathbf { j } + y z ^ { 2 } \mathbf { k }$ . Find the value of the divergence of F at the point $( 0,0,1 )$ .

Evaluate the surface integral $\iint _ { S } z d S$ , where S is that part of the cylinder $z = \sqrt { 1 - x ^ { 2 } }$ that lies above the square with vertices $( - 1 , - 1 ) , ( 1 , - 1 ) , ( - 1,1 )$ , and $( 1,1 )$ .

An upper hemisphere is given by $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ with the density function $\rho ( x , y , z ) = z$ . Find the mass of the sphere.

Find the gradient vector field of the function $f ( x , y ) = x y ^ { 2 }$ .

Evaluate the line integral $\int _ { C } x y ^ { 2 } d x + x ^ { 2 } y d y$ : (a) if $C$ is the curve $x = \cos t , y = \sin t , \quad 0 \leq t \leq \pi$ , the top half of the unit circle from $( 1,0 )$ to $( - 1,0 )$ .(b) if $C$ is the line segment from $( 1,0 )$ to $( - 1,0 )$ .

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

Find the work done by the force field $\mathbf { F } ( x , y ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathbf { i } + \frac { y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathbf { j }$ on a particle that moves along the curve $C$ given in the figure below.

Use Stokes' Theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = x ^ { 2 } z \mathbf { i } + x y ^ { 2 } \mathbf { j } + x ^ { 2 } \mathbf { k }$ and C is the curve of intersection of the plane $x + y + z = 1$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 9$ .

Determine whether or not $\mathbf { F } ( x , y ) = ( y \cos x - \cos y ) \mathbf { i } + ( \sin x + x \sin y ) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$

Sketch the vector field F where F $( x , y ) = \frac { y \mathbf { i } - x \mathbf { j } } { \sqrt { x ^ { 2 } + y ^ { 2 } } }$ .

Evaluate the line integral $\int _ { C } y d x - x d y$ along the circle $x = 2 \cos t , y = 3 \sin t , 0 \leq t \leq 2 \pi$ .

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the triangle with vertices $( 1,0,0 ) , ( 0.1 .0 )$ , and $( 0.0 .1 )$ , oriented counter clockwise as viewed from above.

Determine whether or not $\mathbf { F } ( x , y ) = \left( 3 x ^ { 2 } - 4 y \right) \mathbf { i } + \left( 4 y ^ { 2 } - 2 x \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$ .

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

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 from the point $( 1,0 )$ to the point $( 2,2 )$ along the parabola $y = x ^ { 2 } - x$ .

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Determine whether or not the vector field $\mathbf { F } ( x , y , z ) = z \mathbf { i } + 2 y z \mathbf { j } + \left( x + y ^ { 2 } \right) \mathbf { k }$ is conservative. If it is conservative, find a function f such that $\mathbf { F } = \nabla f$ .