Exam 16: Vector Calculus

Find the mass of the surface $S$ having the given mass density. $S$ is part of the plane $x + 5 y + 9 z = 45$ 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.

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = \left( 4 x - 9 y ^ { 2 } \right) \mathbf { i } + 6 y \mathbf { j }$ in moving a particle in the positive direction once around the triangle with vertices $( 0,0 ) , ( 1,0 )$ , and $( 0,1 )$ .

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

Find the curl of the vector field $\mathbf { F }$ . $\mathbf { F } ( x , y , x ) = 6 y z ^ { 4 } \mathbf { i } + 5 x ^ { 5 } y ^ { 2 } \mathbf { j } + 6 x \mathbf { k }$

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.

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

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

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 $( 2,4 )$ .

Find the value of the constant $c$ such that the vector field $\mathbf { G } ( x , y , x ) = \left( 6 x + 2 y ^ { 3 } + 8 z \right) \mathbf { i } + \left( 6 x ^ { 3 } + 2 y + 4 z \right) \mathbf { j } + ( c z - x ) \mathbf { k }$ is the curl of some vector field $\mathbf { F }$ .

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

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

Use the Divergence Theorem to calculate the surface integral $\iint _ { s } \mathbf { F } \cdot d \mathbf { S }$ ; that is, calculate the flux of $\mathbf { F }$ across $S$ . $\mathbf { F } ( x , y , z ) = x y e ^ { z } \mathbf { i } + x y ^ { 2 } z ^ { 3 } \mathbf { j } - y e ^ { z } \mathbf { k }$ $S$ is the surface of the box bounded by the coordinate planes and the planes $x = 5 , y = 2$ and $z = 1$ . Select the correct answer.

Find a vector representation for the surface. The plane that passes through the point $( 2,4,3 )$ and contains the vectors $3 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$ and $4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k }$

Find the mass of the surface $S$ having the given mass density. $S$ is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9 , z \geq 0$ ; the density at a point $P$ on $S$ is equal to the distance between $P$ and the $x y$ -plane.

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

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

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

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.

Determine whether or not vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = 20 x \mathbf { i } + 8 y \mathbf { j } + 10 z \mathbf { k }$

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