Exam 16: Vector Calculus

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

Determine whether F is conservative. If so, find a function f such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y ) = \left( 6 x ^ { 2 } - 4 x ^ { 3 } y \right) e ^ { - 2 x y } \mathbf { i } - 4 x ^ { 4 } e ^ { - 2 x y } \mathbf { j }$

Let R be a plane region of area A bounded by a piecewise-smooth simple closed curve C. Using Green's Theorem, it can be shown that the centroid of R is $( \bar { x } , \bar { y } )$ , where $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y$ $\bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ Use these results to find the centroid of the given region. The triangle with vertices $( 0,0 )$ , $( 3,0 )$ , and $( 3,4 )$ .

Use a computer algebra system to compute the flux of F across S. S is the surface of the cube cut from the first octant by the planes $x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 }$ $\mathbf { F } ( x , y , z ) = 3 \sin x \cos ^ { 2 } y \mathbf { i } + 3 \sin ^ { 3 } z \cos ^ { 4 } z \mathbf { j } + 3 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }$

Find the gradient vector field of the scalar function F. (That is, find the conservative vector field F for the potential function f of F.) $f ( x , y , z ) = 5 x y ^ { 3 } + 3 y z ^ { 2 }$

A fluid with density $1,330$ flows with velocity $\mathbf { v } = y \mathbf { i } + \mathbf { j } + z \mathbf { k }$ Find the rate of flow upward through the paraboloid $z = 9 - \frac { \left( x ^ { 2 } + y ^ { 2 } \right) } { 3 } , x ^ { 2 } + y ^ { 2 } \leq 36$

Calculate the work done by the force field $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } + 10 z ^ { 2 } \right) \mathbf { i } + \left( y ^ { 2 } + 14 x ^ { 2 } \right) \mathbf { j } + \left( z ^ { 2 } + 12 y ^ { 2 } \right) \mathbf { k }$ when a particle moves under its influence around the edge of the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 16$ that lies in the first octant, in a counterclockwise direction as viewed from above.

Find the area of the part of paraboloid $x = y ^ { 2 } + z ^ { 2 }$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 64$

Find the area of the surface. The part of the paraboloid $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u ^ { 2 } \mathbf { k }$ ; $0 \leq u \leq 7$ , $0 \leq v \leq 2 \pi$

The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus, the vectors in a vector field are tangent to the flow lines. The flow lines of the vector field $\mathbf { F } ( x , y ) = 8 x \mathbf { i } - 32 y \mathbf { j }$ satisfy the differential equations $\frac { d x } { d t } = 8 x$ and $\frac { d y } { d t } = - 32 y .$ Solve these differential equations to find the equations of the family of flow lines.

Find the value of the constant c such that the vector field $\mathrm { G } ( x , y , x ) = \left( 5 x + 9 y ^ { 5 } + 3 z \right) \mathbf { i } + \left( 6 x ^ { 3 } + 2 y + 8 z \right) \mathbf { j } + ( c z - x ) \mathbf { k }$ is the curl of some vector field F.

The plot of a vector field is shown below. A particle is moved $\text { from the point }$ $( - 3,2 )$ $\text { to }$ $( 3,2 )$ . By inspection, determine whether the work done by F on the particle is positive, negative, or zero.

Find an equation of the tangent plane to the parametric surface represented by r at the specified point. $\mathbf { r } ( u , v ) = \left( u ^ { 2 } - v ^ { 2 } \right) \mathbf { i } + u \mathbf { j } + v \mathbf { k }$ ; $( 0,3,3 )$

Use Gauss's Law to find the charge contained in the solid hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 144 , z \geq 0$ , if the electric field is $\mathbf { E } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$

Show that 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 $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 4 x y ^ { 2 } - 10 x y ^ { 2 } \right) \mathbf { i } + \left( 4 x ^ { 2 } y - 10 x ^ { 2 } y \right) \mathbf { j }$ ; $A ( 3,0 )$ and $B ( 3,2 )$

Evaluate the surface integral. S is the part of the cylinder $z ^ { 2 } + y ^ { 2 } = 1$ between the planes $x = 0$ and $x = 1$ in the first octant. $\iint _ { S } 4 \left( x ^ { 2 } y + z \right) d S$

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C. $\oint _ { C } ( 6 x y + 8 \ln ( 1 + x ) ) d x + 4 x ^ { 2 } d y$ , where C is the cardioid $r = 3 + 3 \cos \theta$ .

Find the work done by the force field F on a particle that moves along the curve C. $\mathbf { F } ( x , y ) = ( 5 x + 5 y ) \mathbf { i } + 4 x y \mathbf { j }$ ; $C : \mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } + t ^ { 2 } \mathbf { j }$ , $0 \leq t \leq 1$

Find the curl of $2 x ^ { 2 } z \mathbf { i } + 0 \mathbf { j } + 9 x z ^ { 2 } \mathbf { k }$ .

A particle is moving in a velocity field $\mathrm { V } ( x , y , z ) = 2 y \mathbf { i } + 3 x ^ { 2 } \mathbf { j } + ( z - 3 x ) \mathbf { k }$ At time t = 1 the particle is located at the point (1, 5, 5). a). What is the velocity of the particle at t = 1? b). What is the approximate location of the particle at t = 1.01?