Exam 15: Vector Fields

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cone, $\overrightarrow { \mathbf { r } } ( u , v ) = 6 u \cos v \hat { \mathbf { i } } + 6 u \sin v \hat { \mathbf { j } } + u \hat { \mathbf { k } }$ where $0 \leq u \leq 2$ and $0 \leq v \leq 2 \pi$ .

Determine whether the vector field is conservative. If it is, find a potential function for the vector field. $\overrightarrow { \mathbf { F } } ( x , y ) = 10 x ^ { 9 } y \hat { \mathbf { i } } + x ^ { 10 } { \hat { j } }$

Let S be the oriented upwards. Use a computer surface $z=100-x^{2}-y^{2}, z \geq 0$ algebra system to find the rate of mass flow of a fluid of density $\rho$ through $S$ if the velocity field is given by $\mathbf { F } ( x , y , z ) = 0.5 z k$ .

Write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. $y = \frac { x } { 7 } , 0 \leq x \leq 21 \quad x \text {-axis }$

Find the conservative vector field for the potential function $h ( x , y , z ) = 16 x y \ln ( x + y )$ by finding its gradient.

Identify the surface by eliminating the parameters from the vector-valued function $\mathbf { r } ( u , v ) = 6 \cos v \cos u \mathbf { i } + 6 \cos v \sin u \mathbf { j } + 3 \sin v \mathbf { k }$

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function $\mathbf { r } ( u , v ) = 2 \cos v \cos u \mathbf { i } + 2 \cos v \sin u \mathbf { j } + 3 \sin v \mathbf { k }$ and sketch the graph.

Find the divergence of the vector field. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 10 x ^ { 6 } \hat { \mathbf { i } } - x y ^ { 5 } \hat { \mathbf { j } }$