Exam 16: Vector Calculus

Find (a) the divergence and (b) the curl of the vector field F. $\mathbf { F } ( x , y , z ) = \cos z \mathbf { i } + 5 y \sin 3 z \mathbf { j } + 4 x ^ { 2 } z \mathbf { k }$

Determine whether or not F is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } = ( 8 x + 5 y ) \mathbf { i } + ( 5 x + 12 y ) \mathbf { j }$ .

Set up, but do not evaluate, a double integral for the area of the surface with parametric equations $x = 9 u \cos v , y = 9 u \sin v , z = u ^ { 2 } , 0 \leq u \leq 2,0 \leq v \leq 2 \pi$

Find the divergence of the vector field F. $\mathbf { F } ( x , y , x ) = x z ^ { 4 } \mathbf { i } + 2 x ^ { 4 } z \mathbf { j } - 5 y ^ { 3 } z \mathbf { k }$

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ $\mathbf { F } ( x , y , z ) = 7 x y \mathbf { i } + 7 e ^ { x } \mathbf { j } + 7 x y ^ { 2 } \mathbf { k }$ S consists of the four sides of the pyramid with vertices (0, 0, 0), (3, 0, 0), (0, 0, 3), (3, 0,3) and (0, 3, 0) that lie to the right of the xz-plane, oriented in the direction of the positive y-axis.

A plane lamina with constant density $\rho ( x , y ) = 12$ occupies a region in the xy-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 $( 0,5 )$ .

Suppose that F is an inverse square force field, that is, $\mathbf { F } = \frac { 4 \mathbf { r } } { | \mathbf { r } | ^ { 3 } }$ where $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ Find the work done by F in moving an object from a point $P _ { 1 }$ along a path to a point $P _ { 2 }$ in terms of the distances $d _ { 1 }$ and $d _ { 2 }$ from these points to the origin.

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 10 x y \mathbf { i } + 10 y z \mathbf { j } + 7 x z \mathbf { k }$

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

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = e ^ { - 7 x } \mathbf { i } + e ^ { 4 y } \mathbf { j } + e ^ { 5 x } \mathbf { k }$ and C is the boundary of the part of the plane $8 x + y + 8 z = 8$ in the first octant.

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.

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. $\int _ { C } \sin y d x + x \cos y d y$ C is the ellipse $x ^ { 2 } + x y + y ^ { 2 } = 2$

Use the Divergence Theorem to find the flux of F across S; that is, calculate $\iint _ { S } \mathbf { F } \cdot \mathbf { n } d S$ . $\mathbf { F } ( x , y , z ) = ( 9 x y + \cos z ) \mathbf { i } + ( x - \sin z ) \mathbf { j } + 4 x z \mathbf { k }$ ; S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$

Determine whether or not vector field is conservative. If it is conservative, find a function f such that $\mathbf { F } = \nabla f .$ $\mathbf { F } ( x , y , z ) = 12 x \mathbf { i } + 8 y \mathbf { j } + 2 z \mathbf { k }$

Find an equation in rectangular coordinates, and then identify the surface. $r ( u , v ) = 6 \sin u \mathbf { i } + 3 v \mathbf { j } + 7 \cos u \mathbf { k }$

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$ ; $0 \leq u \leq 2$ , $0 \leq v \leq 2 \pi$

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , that is, find the flux of F across S. $\mathbf { F } ( x , y , z ) = - 5 z \mathbf { i } + 3 y \mathbf { j } - 5 x \mathbf { k }$ ; S is the hemisphere $z = \sqrt { 9 - x ^ { 2 } - y ^ { 2 } }$ ; n points upward.

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. $\mathbf { r } ( u , v ) = u e ^ { v } \mathbf { i } + u v \mathbf { j } + v e ^ { - u } \mathbf { k }$ ; u = ln 9, v = 0

Find the moment of inertia about the z-axis of a thin funnel in the shape of a cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } } , 1 \leq z \leq 4$ if its density function is $\rho ( x , y , z ) = 12 - z$

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. $\mathbf { F } ( x , y ) = x ^ { 5 } y ^ { 6 } \mathbf { i } + y ^ { 5 } x ^ { 6 } \mathbf { j }$ $C : \mathbf { r } ( t ) = \sqrt { t } \mathbf { i } + \left( 1 + t ^ { 3 } \right) \mathbf { j } , 0 \leq t \leq 1$