Exam 13: Vector Calculus

Find the mass of a thin funnel in the shape of a cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } } , 1 \leq z \leq 4$ , is its density function is $\rho ( x , y , z ) = 10 - z$ .

Evaluate $\int _ { c } \left( e ^ { x } + \cos x + y \right) d x + \left( \frac { 1 } { 1 + y ^ { 2 } } + y e ^ { y ^ { 2 } } + x \right) d y$ if C is the path starting at $( 0,0 )$ and going along the line segment from $( 0,0 )$ to $( 1,1 )$ , and then along the line segment from $( 1,1 )$ to $( 2,0 )$ .

According to Green's Theorem, the line integral $\int _ { C } y ^ { 2 } d x + x ^ { 2 } d y$ 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 $f ( x , y )$ .

Evaluate $\int _ { c } x y ^ { 2 } d s$ where the $C$ is the right half of the circle $x ^ { 2 } + y ^ { 2 } = 9$ .

Verify that Stokes' Theorem is true for the vector field $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + \lambda \mathbf { j } + \mathbf { k }$ and the cone $z ^ { 2 } = x ^ { 2 } + y ^ { 2 } , 0 \leq z \leq 1$ , oriented upward.

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the circle $x = 3 \sin t , y = 3 \cos t , z = 2,0 \leq t \leq 2 \pi$ .

Consider the vector field $\mathbf { F } ( x , y , z ) = - \frac { x } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { i } + \frac { y } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { j } + \frac { z } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } \mathbf { k }$ (a) Compute the curl of F.(b) Compute the divergence of F.

Evaluate the line integral $\int _ { C } \left( 2 x y ^ { 2 } + \tan \left( x ^ { 3 } \right) \right) d x + \left( x ^ { 2 } y + 2 y \right) d y$ where C consists of the arc of the parabola $y = x ^ { 2 }$ from $( 0,0 )$ to $( 1,1 )$ , followed by the line segments from $( 1,1 )$ to $( 2,4 )$ , from $( 2,4 )$ to $( 0,4 )$ .

Let $\mathbf { F } ( x , y , z ) = - y \mathbf { j } + x \mathbf { j } + e ^ { z ^ { 2 } } \mathbf { k }$ . Evaluate $\iint _ { S } \operatorname { curl } \mathbf { F } \cdot \mathrm { d } \mathbf { S }$ over the surface S given by $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ , with downward orientation.

For what value of the constant b is the vector field $\mathbf { F } ( x , y , z ) = 2 x \mathbf { i } + z \mathbf { j } + b z \mathbf { k }$ incompressible?

Find the flux of $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } z + \frac { 1 } { \ln ( y + 3 ) } \right) \mathbf { i } + \left( x y z + \frac { 1 } { ( y + 3 ) } \right) \mathbf { j } + \left( x z ^ { 2 } + \frac { z } { ( y + 3 ) ^ { 2 } } \right) \mathbf { k }$ across the surface of the solid $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 1$ .

Find (a) the curl and (b) the divergence of the vector field $\mathbf { F } ( x , y , z ) = y ^ { 2 } z \mathbf { i } - x ^ { 2 } y z \mathbf { k }$ .

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where F $( x , y ) = 2 x y \mathbf { i } + x ^ { 2 } \mathbf { j } + \mathbf { k }$ , and the curve $C$ is given by the vector function $r ( t ) = t ^ { 2 } \mathbf { i } + t ^ { 3 } \mathbf { j } + t \mathbf { k } , \quad 0 \leq t \leq 1$ .

Evaluate $\oint _ { C } 0 d x + 4 x d y$ , if C is the path shown below, starting and ending at $( 1,1 )$ .

Compute the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ if $\mathbf { F } ( x , y , z ) = z ^ { 2 } \mathbf { k }$ and S is the piece of the sphere $z ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ in the second octant $( x \leq 0 , y \geq 0 , z \geq 0 )$ .

Let S be the outwardly-oriented surface of a solid region E where the volume of E is $10 \mathrm { ft } ^ { 3 }$ . If $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $r = | \mathbf { r } |$ , evaluate the surface integral $\iint _ { s } \nabla \left( r ^ { 2 } \right) d \mathbf { S }$ .

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = x ^ { 2 } y \mathbf { i } - 3 x y ^ { 2 } \mathbf { j } + 4 y ^ { 3 } \mathbf { k }$ where S is the part of the elliptic paraboloid $z = x ^ { 2 } + y ^ { 2 } - 9$ that lies below the square $0 \leq x \leq 1 , \quad 0 \leq y \leq 1$ and has downward orientation.

Let $\mathbf { F } ( x , y , z ) = \frac { x \mathbf { i } + y \mathbf { j } + z \mathbf { k } } { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } }$ . Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where C is the curve of intersection of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 2 x$ .

Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ for the vector field $\mathbf { F } ( x , y , z ) = - x \mathbf { i } - y \mathbf { j } + z ^ { 2 } \mathbf { k }$ where S is part of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between the planes z = 1 and z = 2 with upward orientation.

A surface has the shape of the cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ between $z = 0$ and $z = 1$ with the density function $\rho ( x , y , z ) = 1 - z$ . Find the mass of the surface.