Exam 13: Vector Calculus

Evaluate the line integral $\int _ { C } \sin x d x$ where $C$ is the arc of the curve $x = y ^ { 4 }$ from $( 1 , - 1 )$ to $( 1,1 )$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = y ^ { 2 } \mathbf { i } + x \mathbf { j } + z \mathbf { k }$ and S is the part of the surface $z = x + y ^ { 2 }$ that lies above the rectangle $( 0 \leq x \leq 1,0 \leq y \leq 2 )$ and has upward orientation.

If $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $f ( x , y , z ) = \ln | \mathbf { r } |$ , compute $\nabla f$ and determine where $f = 0$ and where $\nabla f = 0$ . What is the domain of f ?

Let $\mathbf { F } ( x , y , z ) = x \mathbf { k }$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is that part of the plane $z = x$ that lies above the square with vertices $( 0,0 ) , ( 1,0 ) , ( 0,1 )$ , and $( 1,1 )$ and has upward orientation.

Let $\mathbf { F } ( x , y , z ) = \left( x + e ^ { y \tan z } \right) \mathrm { i } + 3 \mathrm { xe } ^ { x z } \mathbf { j } + ( \cos y - z ) \mathbf { k }$ and let S be the surface with equation $x ^ { 4 } + y ^ { 4 } + z ^ { 4 } = 1$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Evaluate $\oint _ { C } \left( 12 x ^ { 2 } y ^ { 2 } - 6 y z ^ { 2 } \right) d x + \left( 8 x ^ { 3 } y - 6 x z ^ { 2 } \right) d y - 12 x y z d z$ , where the path C is the curve of intersection of the paraboloid $3 z = x ^ { 2 } + y ^ { 2 }$ with the plane $3 x = 4 y + z = 12$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where S is the boundary surface of the solid sphere $E : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 4$ and $\mathbf { F } = 3 x \mathbf { i } + 4 y \mathbf { j } + 5 z \mathbf { k }$

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + 3 z \mathbf { k }$ , where S is the hemisphere $z = \sqrt { 16 - x ^ { 2 } - y ^ { 2 } }$ with upward orientation.

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = \frac { x } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { i } + \frac { y } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { j } + \frac { z } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { k }$ and S is the sphere $x ^ { 2 } + y ^ { 2 } + ( z - 5 ) ^ { 2 } = 9$ .

Determine whether $\mathbf { F } ( x , y , z ) = ( y , z , x )$ is conservative and if so, find a potential function.

Let $\mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Let $f ( x , y , z ) = e ^ { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } }$ , find curl $( \nabla f )$ at the point $( 1,1,1 )$ .

Is there a vector field G on $\mathbb { R } ^ { 3 }$ such that curl $\mathbf { G } = y z \mathbf { i } + x y z \mathbf { j } + x y \mathbf { k }$ ? Explain.

Evaluate the line integral $\int _ { C } \left( 3 x ^ { 2 } y ^ { 4 } - 6 x y \right) d x + \left( 4 x ^ { 3 } y ^ { 3 } - 3 x ^ { 2 } \right) d y$ , where $C$ is any path from $( 1,2 )$ to $( 2,1 )$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } = ( x , y , z ) = y \mathbf { j } - z \mathbf { k }$ and S is the cube bounded by $x = \pm 1 , \quad y = \pm 1 , \quad z = \pm 1$ .

Evaluate the line integral $\int _ { C } y d x$ , where $C$ is the curve $x = t , y = t ^ { 2 } , 0 \leq t \leq 1$ .

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. $\mathbf { F } ( x , y ) = e ^ { 2 y } \mathbf { i } + \left( 1 + 2 x e ^ { 2 y } \right) \mathbf { j } ; C : \mathbf { r } ( t ) = t e ^ { t } \mathbf { i } + ( 1 + t ) \mathbf { j } ; 0 \leq t \leq 1$

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( e ^ { x ^ { 2 } } + 3 y x ^ { 2 } , \cos y ^ { 2 } + x ^ { 3 } \right)$ and the curve C is given by C : $x = 2 \sin t , y = 2 \cos t , 0 \leq t \leq 2 \pi$ .

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 $C : x = 2 + \sin t , y = 2 + \cos t , 0 \leq t \leq 2 \pi$ .

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + x y \mathbf { j } + z ^ { 2 } \mathbf { k }$ , and the curve $C$ is given by the vector function $\mathbf { r } ( t ) = \sin t \mathbf { i } + \cos t \mathbf { j } + t ^ { 2 } \mathbf { k } , \quad 0 \leq t \leq \frac { \pi } { 2 }$ .