Exam 16: Vector Calculus

Evaluate the line integral $\underset{C}{\int} \mathbf{F} \cdot d \mathbf{r}$ where $\mathbf { F } ( x , y ) = ( x - y ) \mathbf { i } + ( x y ) \mathbf { j }$ and $C$ is the arc of the circle $x ^ { 2 } + y ^ { 2 } = 16$ traversed counterclockwise from $( 4,0 )$ to $( 0 , - 4 )$ . Round your answer to two decimal places.

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

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = \left( 6 x - 7 y ^ { 2 } \right) \mathbf { i } + 3 y \mathbf { j }$ in moving a particle in the positive direction once around the triangle with vertices $( 0,0 ) , ( 1,0 )$ , and $( 0,1 )$ . Select the correct answer.

Find the gradient vector field of $f$ . Select the correct answer. $f ( x , y , z ) = \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } }$

The plot of a vector field is shown below. A particle is moved from the point $( - 3,2 )$ to $( 3,2 )$ . By inspection, determine whether the work done by $\mathbf { F }$ on the particle is positive, negative, or zero.

Evaluate the surface integral. $\iint _ { S } 8 x z d S$ $S$ is the part of the plane $2 x + 2 y + z = 4$ that lies in the first octant.

Evaluate the surface integral where $S$ is the surface with parametric equations $x = 7 u v$ , $y = 6 ( u + v ) , z = 6 ( u - v ) , u ^ { 2 } + v ^ { 2 } = 3$ $\iint _ { S } 10 y z d S$ ,

Evaluate the line integral over the given curve $C$ . $\int _ { C } ( 5 x + 4 y ) d s ; C : \mathbf { r } ( t ) = ( t - 5 ) \mathbf { i } + t \mathbf { j } , 0 \leq t \leq 3$

Find the gradient vector field of the scalar function $f$ . (That is, find the conservative vector field $\mathbf { F }$ for the potential function $f$ of $\mathbf { F }$ .) $f ( x , y , z ) = 7 x y ^ { 3 } + 7 y z ^ { 2 }$

Consider the vector field $\mathbf { F } ( x , y ) = \mathbf { i } + x \mathbf { j }$ If a particle starts at the point $( 10,3 )$ in the velocity field given by $\mathbf { F }$ , find an equation of the path it follows.

Evaluate the surface integral. Round your answer to four decimal places. $\iint _ { S } 3 z d S$ $S$ is surface $x = y ^ { 2 } + 2 z ^ { 2 } , 0 \leq y \leq 1,0 \leq z \leq 1$ .

Calculate the work done by the force field $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } + 10 z ^ { 2 } \right) \mathbf { i } + \left( y ^ { 2 } + 14 x ^ { 2 } \right) \mathbf { j } + \left( z ^ { 2 } + 12 y ^ { 2 } \right) \mathbf { k }$ when a particle moves under its influence around the edge of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies in the first octant, in a counterclockwise direction as viewed from above.

Evaluate $\int _ { C } x y ^ { 4 } d S$ , where $C$ is the right half of the circle $x ^ { 2 } + y ^ { 2 } = 9$ .

Find the area of the part of the surface $y = 20 x + z ^ { 2 }$ that lies between the planes $x = 0 , x = 4 , z = 0$ , and $z = 1$ .

Evaluate the line integral over the given curve $C$ . $\int _ { C } 3 y ^ { 2 } z d s ; C : \mathbf { r } ( t ) = 10 t \mathbf { i } + \sin 7 t \mathbf { j } + \cos 7 t \mathbf { k } , 0 \leq t \leq \frac { \pi } { 2 }$

Find parametric equations for $C$ , if $C$ is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ oriented counterclockwise as viewed from above.

Find the divergence of the vector field $\mathbf { F }$ . $\mathbf { F } ( x , y , x ) = x z ^ { 2 } \mathbf { i } + 4 x ^ { 3 } z \mathbf { j } - 6 y ^ { 4 } z \mathbf { k }$ Select the correct answer.

Below is given the plot of a vector field $\mathbf { F }$ in the $x y$ -plane. (The $z$ -component of $\mathbf { F }$ is 0 .) By studying the plot, determine whether $\operatorname { div } \mathbf { F }$ is positive, negative, or zero.

A plane lamina with constant density $\rho ( x , y ) = 6$ occupies a region in the $x y$ -plane bounded by a simple closed path $C$ . Its moments of inertia about the axes are $I _ { x } = - \frac { \rho } { 3 } \int _ { C } y ^ { 3 } d x \text { and } I _ { y } = \frac { \rho } { 3 } \int _ { C } x ^ { 3 } d y$ Find the moments of inertia about the axes, if $C$ is a rectangle with vertices $( 0,0 ) , ( 4,0 )$ , $( 4,5 )$ and $( 0,5 )$

Evaluate the surface integral where $S$ is the surface with parametric equations $x = 7 u v$ , $y = 6 ( u + v ) , z = 6 ( u - v ) , u ^ { 2 } + v ^ { 2 } = 3$ $\iint _ { S } 10 y z d S$ Select the correct answer.