Exam 13: Vector Calculus

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y ) = x y ^ { 2 } \mathbf { i } + y \mathbf { j }$ and the curve $C$ is the straight line from $( 0,0 )$ to $( 1,1 )$ .

If R is the region enclosed by a simple closed positively-oriented curve C, then the area of R is given by _____. A) $\int _ { C } y d x + x d y$ d. $\frac { 1 } { 2 } \int _ { C } y d x - x d y$ b. $\frac { 1 } { 2 } \int _ { C } x d y - y d x$ e.None of the above C) $\frac { 1 } { 2 } \int _ { C } y d x + x d y$

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

Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Evaluate $\int _ { C } z d x + 2 y z d y + \left( x + y ^ { 2 } \right) d z$ , where $C$ is the line segment $x = 2 - t , y = 2 t , z = 2 t - 1,0 \leq t \leq 1$ .

Let $\mathbf { F } ( x , y , z ) = x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 2 } z ^ { 2 } \mathbf { j } + y ^ { 2 } z ^ { 2 } \mathbf { k }$ . Let C be the rectangular path from $( 1,1,2 )$ to $( 3,1,2 )$ to $( 3,5,2 )$ to $( 1,5,2 )$ to $( 1,1,2 )$ . Use Stokes' Theorem to evaluate the line integral $\int _ { C } \mathbf { F } \cdot \mathbf { T } d \mathrm {~s}$ , where T is the unit tangent vector to C.

Sketch the vector field F where $\mathbf { F } ( x , y , z ) = \mathbf { j } + \mathbf { k }$ .

Consider the surfaces $S _ { 1 }$ : $\frac { 1 } { 9 } x ^ { 2 } + \frac { 1 } { 9 } y ^ { 2 } + \frac { 1 } { 4 } z ^ { 2 } s = 1 , z \geq 0$ , and $S _ { 2 }$ : $4 z = 9 - x ^ { 2 } - y ^ { 2 } , \quad z \geq 0$ , and let F be a vector field with continuous partial derivatives everywhere. Why do we know that $\iint _ { S _ { 1 } } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S } = \iint _ { S _ { 2 } } \operatorname { curl } \mathbf { F } \cdot d \mathbf { S }$ ?

Evaluate the line integral $\int _ { C } y d x + x d y$ , where $C$ is the curve $x = t , y = t ^ { 3 } , 1 \leq t \leq 2$ .

Use Green's Theorem to evaluate the line integral along the given positively oriented curve: $\int _ { C } x ^ { 2 } y d x - 3 y ^ { 2 } d y$ , where C is the circle $x ^ { 2 } + y ^ { 2 } = 1$ .

According to Green's Theorem, the line integral $\int _ { C } y d x + x d y$ over a positively oriented, piecewise-smooth, simple closed curve C is equal to the double integral $\iint _ { D } f ( x , y ) d A$ over the region D bounded by C. Find the function $f ( x , y )$ .

A fluid has density 1500 and velocity field $\mathbf { v } = - y \mathbf { i } + x \mathbf { j } + 2 z \mathbf { k }$ . Find the rate of flow outward through the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ .

Evaluate the line integral $\int _ { c } x y d s$ where C is the line segment joining $( - 1,1 )$ to $( 2,3 )$ .

Find the flux of $\mathbf { F } ( x , y , z ) = \left( x + y z ^ { 2 } \right) \mathbf { i } + \left( x z ^ { 2 } + y \right) \mathbf { j } + \left( z + y x ^ { 2 } \right) \mathbf { k }$ across the surface of the solid bounded by the paraboloid $y = 4 - z ^ { 2 } - x ^ { 2 }$ and the $x z -$ plane.

Evaluate the line integral $\oint _ { C } x y ^ { 2 } d x - y x ^ { 2 } d y$ around the triangle with vertices $( 1,0 )$ , $( 0,1 )$ , and $( 0,0 )$ with clockwise orientation.

Let $\mathbf { F } ( x , y , z ) = x y ^ { 2 } \mathbf { i }$ and let S be the boundary surface of the solid $E = \{ ( x , y , z ) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Consider the top half of the ellipsoid $x ^ { 2 } + y ^ { 2 } + \frac { 1 } { 4 } z ^ { 2 } = 1 , z \geq 0$ parametrized by $x = \sin u \sin v , y = \cos u \sin v , z = 2 \cos v , 0 \leq v \leq \frac { \pi } { 2 }$ . Find a normal vector N at the point determined by $u = \frac { \pi } { 4 } , v = \frac { \pi } { 3 }$ , and determine if it is upward and/or outward.

Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y z \right) \mathbf { i } + x ^ { 2 } y \mathbf { j } + x z ^ { 2 } \mathbf { k }$ and let S be the surface of the solid bounded by the spheres $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ and $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Evaluate the line integral $\int _ { C } x d y$ 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 $( 0,0 )$ .

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