Exam 13: Vector Calculus

Evaluate the surface integral $\iint _ { S } x z d S$ , where S is the triangle with vertices $( 1,0,0 ) , ( 0.1 .0 )$ , and $( 0,0,1 )$

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ and the curve $C$ is given by $C : x = 5 \sin t , y = 5 \cos t , 0 \leq t \leq 2 \pi$ .

Find a function of $f ( x , y )$ such that $\nabla f = \mathrm { F } ( x , y ) = \left\langle x y ^ { 2 } + 2 , x ^ { 2 } y + 2 \right\rangle$ .

Find the work done by the force field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ on a particle that moves along the quarter-circle $C$ given in the figure below.

Find the value of the Laplacian $\nabla ^ { 2 } f \text {of } f ( x , y ) = x y ^ { 3 }$ at the point $( 1,1 )$

Use the Divergence Theorem to evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = x ( y + 1 ) \mathbf { i } + 2 y z \mathbf { j } - \left( z ^ { 2 } + y z \right) \mathbf { k }$ and S is the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ .

Evaluate $\oint _ { C } \left( \frac { 1 } { 4 } y ^ { 4 } - \frac { 1 } { 2 } x ^ { 3 } \cos y ^ { 3 } \right) d x + \left( x y ^ { 3 } + x ^ { 2 } + \frac { 3 } { 8 } x ^ { 4 } y ^ { 2 } \sin y ^ { 3 } \right) d y$ , if C is the path shown below.

Evaluate the line integral $\int _ { c } x d x$ , where $C$ is the curve $x = t , \quad y = t , \quad 0 \leq t \leq 1$ .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $\mathbf { F } ( x , y ) = x ^ { 3 } y \mathbf { i } + x ^ { 4 } \mathbf { j }$ and C is the curve $x ^ { 4 } + y ^ { 4 } = 1$ .

Let S be the parametric surface $x = r \cos \theta , y = r \sin \theta , z = \theta , 0 \leq r \leq 1,0 \leq \theta \leq \frac { \pi } { 2 }$ . Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + z \mathbf { k }$ .

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 curve C. $\mathbf { F } ( x , y , z ) = 4 x e ^ { x } \mathbf { i } + \cos y \mathbf { j } + 2 x ^ { 2 } \mathrm { e } ^ { z } \mathbf { k } ; C : \mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + t ^ { 4 } \mathbf { k } , \quad 0 \leq t \leq 1$

Determine whether or not $\mathbf { F } ( x , y ) = \left( y e ^ { x y } + 4 x ^ { 3 } y \right) \mathbf { i } + \left( x e ^ { x y } + x ^ { 4 } \right) \mathbf { j }$ is a conservative vector field. If it is, find a function f such that $\mathbf { F } = \nabla f$ .

Find the value of the Laplacian $\nabla ^ { 2 } f \text { of } f ( x , y , z ) = z \left( x ^ { 2 } + y ^ { 3 } \right)$ at the point $( 1,2,3 )$

Let $\mathbf { F } ( x , y , z ) = x \mathbf { j }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ along the rectangular path from $( 0,0,0 )$ to $( 1,0,1 )$ to $( 1,1,1 )$ to $( 0,1,0 )$ to $( 0,0,0 )$ .

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = x \mathbf { i } + y \mathbf { j } + x \mathbf { k }$ and S is the part of the plane $z = 1 - x - y$ in the first octant with downward orientation.

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } \left( y ^ { 2 } - \tan ^ { - 1 } x \right) d x + ( 3 x + \sin y ) d y$ , where C is the boundary of the region enclosed by the parabola $y = x ^ { 2 }$ and the line $y = 4$ .

Find the work done by the force field $\mathbf { F } ( x , y ) = x y ^ { 2 } \mathbf { i } + y \mathbf { j }$ on a particle that moves along the curve $y = x ^ { 2 }$ from $( 0,0 )$ to $( 1.1 )$ .

Evaluate $\iint _ { S } \frac { x ^ { 2 } + z } { x ^ { 2 } + y ^ { 2 } } d S$ , where S is the part of the surface $z = x y$ that lies between the cylinders $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 4$ .

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = y ^ { 2 } z \mathbf { i } + x z \mathbf { j } + x ^ { 2 } y ^ { 2 } \mathbf { k }$ and S is the part of the paraboloid $z = x ^ { 2 } + y ^ { 2 }$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ , oriented upward.

Find a function of $f ( x , y )$ such that $\nabla f = 2 x y \mathbf { i } + x ^ { 2 } \mathbf { j }$ .