Exam 16: Vector Calculus

Find the work done by the force field $\mathbf { F } ( x , y ) = 24 x \mathbf { i } + 12 ( 2 y + 1 ) \mathbf { j }$ in moving an object along an arch of the cycloid $\mathbf { r } ( t ) = ( t - \sin ( t ) ) \mathbf { i } + ( 1 - \cos ( t ) ) \mathbf { j } , \quad 0 \leq t \leq 2 \pi$ .

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 $( 0,1 )$ .

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 $z = 2$ .

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

Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 21 x ^ { 2 } y \mathbf { i } + \left( 7 x ^ { 3 } + 15 y ^ { 2 } z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 3 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0 , - 3 )$

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.

Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 27 x ^ { 2 } y \mathbf { i } + \left( 9 x ^ { 3 } + 24 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 12 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0,1 )$

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

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 $4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k }$

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

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

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 $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 36$

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

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$ . $\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y$ , where $\mathrm { C }$ is the triangle with vertices $( 0,0 ) , ( 3,4 )$ , and $( 0,4 )$

Find the area of the surface $S$ where $S$ is the part of the surface $x = y z$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 16$ .

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = ( x - 9 z ) \mathbf { i } + ( 7 x + y + 2 z ) \mathbf { j } + ( x - 16 y ) \mathbf { k }$

Let $f$ be a scalar field. Determine whether the expression is meaningful. If so, state whether the expression represents a scalar field or a vector field. $\nabla ( \operatorname { grad } f )$

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 $( 0,10 )$ .