Exam 16: Vector Calculus

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

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r } , \mathbf { F } ( x , y , z ) = 6 y x ^ { 2 } \mathbf { i } + 2 x ^ { 3 } \mathbf { j } + 6 x y \mathbf { k }$ , $C$ is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ oriented counterclockwise as viewed from above.

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.) $\mathbf { F } ( x , y ) = \left( 18 x ^ { 2 } y ^ { 2 } - 5 y \sin x \right) \mathbf { i } + \left( 12 x ^ { 3 } y + 5 \cos x \right) \mathbf { j }$ $C : \mathbf { r } ( t ) = ( - 2 - \sin t ) \mathbf { i } + 2 \cos t \mathbf { j } ; \quad 0 \leq t \leq \pi$

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.

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

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$ . $\oint _ { C } ( 6 x y + 5 \ln ( 1 + x ) ) d x + 2 x ^ { 2 } d y$ , where $C$ is the cardioid $r = 2 + 2 \cos \theta$ .

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

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = z ; S$ is the part of the torus with vector representation $\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin u \mathbf { j } + 3 \sin v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 }$

Evaluate the line integral over the given curve C. $\int _ { C } 4 x y d s , \text { where } C \text { is the line segment joining } ( - 2 , - 1 ) \text { to } ( 4,5 )$

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

Evaluate $\int _ { C } y z d y + x y d z$ , where $C$ is given by $x = 10 \sqrt { t } , y = 3 t , z = 10 t ^ { 2 } , 0 \leq t \leq 1$ . Round your answer to two decimal place.

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.

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 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.

Find parametric equations for $C$ , if $C$ is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ oriented counterclockwise as viewed from above.

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 a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 10 z ^ { 2 } = 5$ that lies in front of the plane $x = 0$ .

Find the correct identity, if $f$ is a scalar field, $\mathbf { F }$ and $\mathbf { G }$ are vector fields.