Exam 16: Vector Calculus

Show that $\mathbf { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 21 x ^ { 2 } y \mathbf { i } + \left( 7 x ^ { 3 } + 15 y ^ { 2 } z ^ { 2 } \right) \mathbf { j } + 10 y ^ { 3 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0 , - 3 )$

Find the area of the surface $S$ where $S$ is the part of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 9$ that lies to the right of the $x z$ -plane and inside the cylinder $x ^ { 2 } + z ^ { 2 } = 4$ .

Find the moment of inertia about the $z$ -axis of a thin funnel in the shape of a cone $z = \sqrt { x ^ { 2 } + y ^ { 2 } } , 1 \leq z \leq 4$ , if its density function is $\rho ( x , y , z ) = 14 - z$ .

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$ . $\oint _ { C } ( 6 x y + 5 \ln ( 1 + x ) ) d x + 2 x ^ { 2 } d y$ , where $C$ is the cardioid $r = 2 + 2 \cos \theta$ .

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $r = | \mathbf { r } |$ . Find $\nabla \cdot ( \mathbf { r } )$

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = ( 2 \sinh 2 z ) \mathbf { i } + \left( 4 e ^ { 4 z } \cos 4 y \right) \mathbf { j } + ( 4 x \cosh 2 z ) \mathbf { k }$

Use Stokes' Theorem to evaluate $\oint _ { C } \mathbf { F } \cdot d \mathbf { r }$ . $\mathbf { F } ( x , y , z ) = 7 \cos x \mathbf { i } + 5 e ^ { y } \mathbf { j } + 2 x y \mathbf { k }$ $C$ is the curve obtained by intersecting the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ with the hyperbolic paraboloid $z = x ^ { 2 } - y ^ { 2 }$ , oriented in a counterclockwise direction when viewed from above

Let $\mathbf { F } = \nabla f$ , where $f ( x , y ) = \sin ( x - 6 y )$ Which of the following equations does the line segment from $( 0,0 )$ to $( 0 , \pi )$ satisy? Select the correct answer.

Find a parametric representation for the part of the elliptic paraboloid $x + y ^ { 2 } + 10 z ^ { 2 } = 5$ that lies in front of the plane $x = 0$ .

Find the work done by the force field $\mathbf { F }$ on a particle that moves along the curve $C$ . $\mathbf { F } ( x , y ) = ( 5 x + 5 y ) \mathbf { i } + 5 x y \mathbf { j } ; \quad C : \mathbf { r } ( t ) = 2 t ^ { 2 } \mathbf { i } + t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ . Select the correct answer. $\mathbf { F } ( x , y ) = \left( 5 x ^ { 4 } + 4 y \right) \mathbf { i } + \left( 7 x - 5 y ^ { 4 } \right) \mathbf { j }$

Find a parametric representation for the part of the plane $z = 6$ that lies inside the cylinder $x ^ { 2 } + y ^ { 2 } = 49$ .

A thin wire is bent into the shape of a semicircle $x ^ { 2 } + y ^ { 2 } = 4 , x > 0$ . If the linear density is 4 , find the exact mass of the wire. Select the correct answer.

Determine whether or not vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = 5 z y \mathbf { i } + 5 x z \mathbf { j } + 5 x y \mathbf { k }$

Let $S$ be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

A particle starts at the point $( - 3,0 )$ , moves along the $x$ -axis to $( 3,0 )$ and then along the semicircle $y = \sqrt { 9 - x ^ { 2 } }$ to the starting point. Use Green's Theorem to find the work done on this particle by the force field $\mathbf { F } ( x , y ) = \left\{ 24 x , 8 x ^ { 3 } + 24 x y ^ { 2 } \right\}$ Select the correct answer.

Match the equation with one of the graphs below. $\mathbf { r } ( u , v ) = 3 \sin u \cos v \mathbf { i } + 3 \sin u \sin v \mathbf { j } + 3 \cos u \mathbf { k }$

Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 27 x ^ { 2 } y \mathbf { i } + \left( 9 x ^ { 3 } + 24 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 12 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( - 1,0,1 )$