Exam 16: Vector Calculus

Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$ . $\mathbf { F } ( x , y , z ) = 7 z \mathbf { i } + y \mathbf { j } + 4 x z \mathbf { k }$ ; C is the boundary of the triangle with vertices $( 6,0,0 )$ , $( 0,6,0 )$ , and $( 0,0,6 )$ oriented in a counterclockwise direction when viewed from above

Consider the vector field $\mathbf { F } ( x , y ) = \mathbf { i } + x \mathbf { j }$ If a particle starts at the point $( 10,4 )$ in the velocity field given by F, find an equation of the path it follows.

Find the exact mass of a thin wire in the shape of the helix $x = 2 \sin ( t ) , y = 2 \cos ( t ) , z = 3 t , 0 \leq t \leq 2 \pi$ if the density is 5.

Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the vector field F and the path C. (Hint: Show that F is conservative, and pick a simpler path.) $\mathbf { F } ( x , y ) = \left( 16 x ^ { 3 } y ^ { 2 } - 4 y \sin x \right) \mathbf { i } + \left( 8 x ^ { 4 } y + 4 \cos x \right) \mathbf { j }$ C: $\mathbf { r } ( t ) = ( 3 - \sin t ) \mathbf { i } + 2 \cos t \mathbf { j } ; \quad 0 \leq t \leq \pi$

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.

Determine whether F is conservative. If so, find a function f such that $\mathbf { F } = \nabla f .$ . $\mathbf { F } ( x , y , z ) = 9 x ^ { 2 } y ^ { 4 } z ^ { 2 } \mathbf { i } + 12 x ^ { 3 } y ^ { 3 } z ^ { 2 } \mathbf { j } + 6 x ^ { 3 } y ^ { 4 } z \mathbf { k }$

Determine whether or not F is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } = \left( 8 y e ^ { 8 x } + \sin y \right) \mathbf { i } + \left( e ^ { 8 x } + x \cos y \right) \mathbf { j }$

A thin wire in the shape of a quarter-circle $\mathbf { r } ( t ) = 6 \cos t \mathbf { i } + 6 \sin t \mathbf { j }$ , $0 \leq t \leq \frac { \pi } { 2 }$ , has a linear mass density $\pi ( x , y ) = 2 x + 4 y$ . Find the mass and the location of the center of mass of the wire.

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = 5 x + 6 y + z$ ; S is the part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between the planes $z = 1$ and $z = 4$ .

Find the work done by the force field $\mathbf { F } ( x , y ) = 24 x \mathbf { i } + 12 ( 2 y + 1 ) \mathbf { j }$ in moving an object along an arch of the cycloid $\mathbf { r } ( t ) = ( t - \sin ( t ) ) \mathbf { i } + ( 1 - \cos ( t ) ) \mathbf { j } , 0 \leq t \leq 2 \pi$

Use Green's Theorem and/or a computer algebra system to evaluate $\int _ { C } x ^ { 2 } y d x - x y ^ { 2 } d y$ where C is the circle $x ^ { 2 } + y ^ { 2 } = 64$ with counterclockwise orientation.

Find the curl of the vector field F. $\mathbf { F } ( x , y , x ) = 3 y z ^ { 3 } \mathbf { i } + 8 x ^ { 5 } y ^ { 4 } \mathbf { j } + 4 x \mathbf { k }$

A particle starts at the point $( - 3,0 )$ , moves along the x-axis to (3, 0) and then along the semicircle $y = \sqrt { 9 - x ^ { 2 } }$ to the starting point. Use Green's Theorem to find the work done on this particle by the force field $\mathbf { F } ( x , y ) = \left\langle 24 x , 8 x ^ { 3 } + 24 x y ^ { 2 } \right\rangle$

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 given curve C. $\mathbf { F } ( x , y ) = y \mathbf { i } + ( x + 6 y ) \mathbf { j }$ C is the upper semicircle that starts at (1, 2) and ends at (5, 2).

Find the correct identity, if f is a scalar field, F and G are vector fields.

Match the equation with one of the graphs below. $\mathbf { r } ( u , v ) = u ^ { 2 } \mathbf { i } + u \cos v \mathbf { j } + u \sin v \mathbf { k }$

Let $\mathbf { F } = \nabla f ,$ where $f ( x , y ) = \sin ( x - 8 y )$ . Which of the following equations does the line segment from $( 0,0 )$ to $( 0 , \pi )$ satisy?

Find the div F if $\mathbf { F } ( x , y , z ) = e ^ { x z } ( \cos y z \mathbf { i } + \sin y z \mathbf { j } - \mathbf { k } )$ .

Below is given the plot of a vector field F in the xy-plane. (The z-component of F is 0.) By studying the plot, determine whether div F is positive, negative, or zero.

Let F be a vector field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. $\nabla \cdot ( \nabla \times \mathbf { F } )$