Exam 16: Vector Calculus

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = \left( 6 x - 7 y ^ { 2 } \right) \mathbf { i } + 3 y \mathbf { j }$ in moving a particle in the positive direction once around the triangle with vertices $( 0,0 ) , ( 1,0 )$ , and $( 0,1 )$ .

Let $D$ be a region bounded by a simple closed path $C$ in the $x y$ . Then the coordinates of the centroid $( \bar { x } , \bar { y } )$ of $D$ are $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where $A$ is the area of $D$ . Find the centroid of the triangle with vertices $( 0,0 ) , ( 8,0 )$ and $( 0,16 )$ .

Let $D$ be a region bounded by a simple closed path $C$ in the $x y$ . Then the coordinates of the centroid $( \bar { x } , \bar { y } )$ of $D$ are $\bar { x } = \frac { 1 } { 2 A } \oint _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \oint _ { C } y ^ { 2 } d x$ where $A$ is the area of $D$ . Find the centroid of the triangle with vertices $( 0,0 ) , ( 5,0 )$ and $( 0,10 )$ .

Find a function $f$ such that $\mathbf { F } = \nabla f$ , and use it to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the given curve $C$ . (x,y)=+ C:(t)=+ 1+ ,0\leqt\leq1

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

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$ . $\oint _ { C } 5 x y d x + 4 x ^ { 2 } d y$ , where $C$ is the triangle with vertices $( 0,0 ) , ( 5,4 )$ , and $( 0,4 )$ .

Suppose that $f ( x , y , z ) = g \left( \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \right)$ where $g$ is a function of one variable such that $g ( 6 ) = 2$ . Evaluate $\iint _ { S } f ( x , y , z ) d S$ where $S$ is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$

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

Find the exact mass of a thin wire in the shape of the helix $x = 2 \sin ( t ) , y = 2 \cos ( t ) , z = 5 t , 0 \leq t \leq 2 \pi$ if the density is 5 . 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 .

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

A plane lamina with constant density $\rho ( x , y ) = 12$ occupies a region in the $x y$ -plane bounded by a simple closed path $C$ . Its moments of inertia about the axes are $I _ { x } = - \frac { \rho } { 3 } \int _ { C } y ^ { 3 } d x \text { and } l _ { y } = \frac { \rho } { 3 } \int _ { C } x ^ { 3 } d y \text {. }$ Find the moments of inertia about the axes, if $C$ is a rectangle with vertices $( 0,0 ) , ( 4,0 )$ , $( 4,5 )$ and $( 0,5 )$ . Select the correct answer.

Evaluate $\iint _ { S } f ( x , y , z ) d S$ . $f ( x , y , z ) = 2 x + 2 y + 5 z ; S$ is the part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between the planes $z = 1$ and $z = 2$ .

Evaluate the line integral over the given curve $C$ . $\int _ { C } 2 y ^ { 2 } z d s ; C : \mathbf { r } ( t ) = 10 t \mathbf { i } + \sin 7 t \mathbf { j } + \cos 7 t \mathbf { k } , 0 \leq t \leq \frac { \pi } { 2 }$

Suppose that $f ( x , y , z ) = g \left( \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \right)$ where $g$ is a function of one variable such that $g ( 3 ) = 4$ Evaluate $\iint _ { S } f ( x , y , z ) d S$ where $S$ is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Select the correct answer.

Evaluate the surface integral. $\iint _ { S } 8 x z d S$ $S$ is the part of the plane $2 x + 2 y + z = 4$ that lies in the first octant. Select the correct answer.

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 8 e ^ { x } \sin ( y ) \mathbf { i } + 4 e ^ { x } \cos ( y ) \mathbf { j } + 3 z \mathbf { k }$

Show that $\mathbf { F }$ 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 $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 15 x ^ { 2 } y ^ { 2 } - 14 x y ^ { 4 } \right) \mathbf { i } + \left( 10 x ^ { 3 } y - 28 x ^ { 2 } y ^ { 3 } \right) \mathbf { j } ; A ( 1 , - 2 )$ and $B ( 1 , - 1 )$

Find the work done by the force field $\mathbf { F } ( x , y ) = 30 x \mathbf { i } + 15 ( 2 y + 1 ) \mathbf { j }$ in moving an object along an arch of the cycloid $\mathbf { r } ( t ) = ( t - \sin ( t ) ) \mathbf { i } + ( 1 - \cos ( t ) ) \mathbf { j } , 0 \leq t \leq 2 \pi$

Find (a) the divergence and (b) the curl of the vector field $\mathbf { F }$ . $\mathbf { F } ( x , y , z ) = \cos z \mathbf { i } + 3 y \sin 3 z \mathbf { j } + 6 x ^ { 3 } z \mathbf { k }$