Exam 16: Vector Calculus

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r } \text {, }$ 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.

Find the area of the surface $S$ where $S$ is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ that lies to the right of the $x z$ -plane and inside the cylinder $x ^ { 2 } + z ^ { 2 } = 4$ .

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

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

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 .

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

Determine whether or not vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = 5 z y \mathbf { i } + 5 x z \mathbf { j } + 5 x y \mathbf { k }$

A plane lamina with constant density $\rho ( x , y ) = 12$ 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 line integral over the given curve $C$ . $\int _ { C } \left( 6 x + 2 y ^ { 2 } \right) d s ; C : \mathbf { r } ( t ) = ( t - 6 ) \mathbf { i } + t \mathbf { j } , 0 \leq t \leq 1$

Assuming that S satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find $\iint _ { s } 4 \mathbf { a } \cdot \mathbf { n } d S \text {, }$ where a is the constant vector.

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $r = | \mathbf { r } |$ . Find $\nabla \cdot ( \mathbf { r } )$ .

Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$ . $\mathbf { F } ( x , y , z ) = 7 \cos x \mathbf { i } + 5 e ^ { y } \mathbf { j } + 2 x y \mathbf { k }$ $C$ is the curve obtained by intersecting the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ with the hyperbolic paraboloid $z = x ^ { 2 } - y ^ { 2 }$ , oriented in a counterclockwise direction when viewed from above

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 $( 0,5 )$ .

____ Show that $\mathbf { F }$ is conservative and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 15 x ^ { 2 } y ^ { 2 } - 14 x y ^ { 4 } \right) \mathbf { i } + \left( 10 x ^ { 3 } y - 28 x ^ { 2 } y ^ { 3 } \right) \mathbf { j } ; A ( 1 , - 2 )$ and $B ( 1 , - 1 )$

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 $\mathbf { F }$ .) $f ( x , y , z ) = 7 x y ^ { 2 } + 2 y z ^ { 2 }$

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.

Determine whether or not vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = 18 x \mathbf { i } + 10 y \mathbf { j } + 4 z \mathbf { k }$

Find the curl of the vector field F. $\mathbf { F } ( x , y , x ) = 7 y z ^ { 2 } \mathbf { i } + 4 x ^ { 5 } y ^ { 5 } \mathbf { j } + 8 x \mathbf { k }$

Which plot illustrates the vector field $\mathbf { F } ( x , y , z ) = \mathbf { k }$ ?