Exam 16: Vector Calculus

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. Select the correct answer.

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 $( 0,5 ) .$

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $r = | \mathbf { r } |$ . Find $\nabla \cdot ( 6 \mathbf { r } )$ Select the correct answer.

Find a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 8 z ^ { 2 } = 9$ that lies in front of the plane $x = 0$ . Select the correct answer.

Evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ for the vector field $\mathbf { F }$ and the path $C$ . (Hint: Show that $\mathbf { F }$ is conservative, and pick a simpler path.) (x,y)= 18-5yx + 12y+5x C:(t)=(-2-t)+2t;0\leqt\leq\pi

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k } ; 0 \leq u \leq 3,0 \leq v \leq 2 \pi$

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

Let $S$ be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares. Select the correct answer.

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 work done by the force field $\mathbf { F } ( x , y ) = x \sin ( y ) \mathbf { i } + y \mathbf { j }$ on a particle that moves along the parabola $y = x ^ { 2 }$ from $( 1,1 )$ to $( 2,4 )$ .

Evaluate $\iint _ { S } f ( x , y , z ) d S$ $f ( x , y , z ) = x + y ; S$ is the part of the plane $2 x + 4 y + 3 z = 24$ in the first octant.

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ . $\mathbf { F } ( x , y , z ) = 6 x y \mathbf { i } + 5 y z \mathbf { j } + 2 z ^ { 2 } \mathbf { k }$ $S$ is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ lying below the plane $z = 6$ and oriented with normal pointing downward.

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y ) = \left( 14 x - 14 x ^ { 2 } y \right) e ^ { - 2 x y } \mathbf { i } - 14 x ^ { 3 } e ^ { - 2 x y } \mathbf { j }$

Use Stoke's theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d r$ . $\mathbf { F } ( x , y , z ) = 4 z \mathbf { i } + 2 x \mathbf { j } + 6 y \mathbf { k }$ $C$ is the curve of intersection of the plane $z = x + 9$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 9$

Find the mass of the surface $S$ having the given mass density. $S$ is part of the plane $x + 7 y + 3 z = 21$ in the first octant; the density at a point $P$ on $S$ is equal to the square of the distance between $P$ and the $x y$ -plane.

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

Match the vector field with its plot. $\mathbf { F } ( x , y ) = \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$

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 $\iint _ { s } \mathbf { a } \cdot \mathbf { n } d S \text {, }$ where a is the constant vector.