Exam 13: Vector Calculus

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 gradient vector field of $f ( x , y ) = \sin ( 2 x + 3 y )$ .

Evaluate the line integral $\int _ { C } x e ^ { x ^ { 2 } } d x + y \sin y ^ { 2 } d y$ along the triangular path consisting of the line segment from $( 0,0 )$ to $( 2,3 )$ followed by the line segment from $( 2,3 )$ to $( 1,0 )$ followed by the line segment from $( 1,0 )$ to $( 0,0 )$ .

Find (a) the curl and (b) the divergence of the vector field $\mathbf { F } ( x , y , z ) = ( x + 3 y - 5 z ) \mathbf { i } + ( z - 3 y ) \mathbf { j } + ( 5 x + 6 y - z ) \mathbf { k }$ .

Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } y \mathbf { i } - x ^ { 2 } z \mathbf { j } + z ^ { 2 } y \mathbf { k }$ and let S be the surface of the rectangular box bounded by the planes $x = 0 , x = 3 , y = 0 , y = 2 , z = 0$ , and $z = 1$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Evaluate the line integral by two methods: directly and using Green's Theorem. $\oint _ { C } x d x - x ^ { 2 } y ^ { 2 } d y$ , where C is the triangle with vertices $( 0,0 )$ , $( 1,1 )$ , and $( 0,1 )$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot \mathbf { n } d \mathbf { S }$ , where S is the cube bounded by the planes $x = \pm 1 , y = \pm 1$ and $z = \pm 1 , \mathbf { F } = x ^ { 2 } y \mathbf { i } + x y \mathbf { j } + y ^ { 2 } z ^ { 3 } \mathbf { k }$ , and n is the outward normal.

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ where $\mathbf { F } ( x , y , z ) = \left( x + \tan ^ { - 1 } y z \right) \mathbf { i } + y ^ { 2 } z \mathbf { j } + z \mathbf { k }$ and S is the part of the hemisphere $z = \sqrt { 9 - y ^ { 2 } - z ^ { 2 } }$ that lies inside the cylinder $y ^ { 2 } + z ^ { 2 } = 4$ , oriented in the direction of the positive x-axis.

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } \left( x ^ { 3 } - y ^ { 3 } \right) d x + \left( x ^ { 3 } + y ^ { 3 } \right) d y$ , where C is the boundary of the region between the circles $x ^ { 2 } + y ^ { 2 } = 1$ and $x ^ { 2 } + y ^ { 2 } = 9$ .

Evaluate the flux of the vector field $\mathbf { F } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }$ through the plane region with the given orientation as shown below.

Let $\mathbf { F } ( x , y , z ) = x y \mathbf { j } + x y z \mathbf { k }$ find $\nabla \times ( \nabla \times F )$ .

Find a function of $f ( x , y )$ such that $\nabla f = \mathrm { F } ( x , y ) = \left( \frac { y } { 1 + x ^ { 2 } y ^ { 2 } } , \frac { x } { 1 + x ^ { 2 } y ^ { 2 } } \right)$ .

For what value of the constant b is the vector field $\mathbf { F } = b x y ^ { 2 } \mathbf { i } + x ^ { 2 } y \mathbf { j }$ irrotational?

Let C be the closed path from $( 0,0 )$ to $( 2,4 )$ along $y = x ^ { 2 }$ and back again from $( 2,4 )$ to $( 0,0 )$ along $y = 2 x$ . Evaluate $\int _ { C } \left( x ^ { 3 } + 2 y \right) d x + \left( x - y ^ { 2 } \right) d y$ directly, and then using Green's Theorem.

Evaluate $\int _ { C } 15 x ^ { 2 } y z d s$ where the $C$ is given by $C : x = t , y = t ^ { 2 } , \quad z = \frac { 2 } { 3 } t ^ { 3 } , \quad 0 \leq t \leq 1$ .

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

Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y \sin z \right) \mathbf { i } + \left( y ^ { 3 } + z ^ { 2 } \sin z \right) \mathbf { j } + \left( z ^ { 3 } + x \right) \mathbf { k }$ and let S be the boundary surface of the solid E bounded by $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } , z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ , and $z = 0$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Evaluate the line integral $\int _ { C } ( x + y ) d x + x ^ { 2 } d y$ , where C is the arc of the parabola $y = x ^ { 2 } - x$ from the point $( 1,0 )$ to the point $( 2,2 )$ .

Evaluate the line integral $\int _ { C } x y d x + ( x - y ) d y$ , where C is the arc of the circle $x ^ { 2 } + y ^ { 2 } = 1$ from the point $( 1,0 )$ to the point $( 0,1 )$ , followed by the line segment from the point $( 0,1 )$ to the point $( 4,3 )$ .

Find the gradient vector field of the function $f ( x , y ) = \ln \left( \sqrt { x ^ { 2 } + y ^ { 2 } } \right)$ .