Exam 20: The Curl and Stokes Theorem

Use the Divergence Theorem to find the flux of the vector field ${ \vec { F } } = \left( y ^ { 2 } x - 3 z ^ { 2 } y \right) \vec { i } + \left( z ^ { 2 } y + z x ^ { 3 } \right) \vec { j } + \left( x ^ { 2 } z + y x \right) \vec { k }$ through the sphere x² + y² + z² = $1$ .

Suppose that curl $\vec {F }$ is not zero.True or false? If curl $\vec { F }$ is parallel to the $\mathcal { X }$ -axis for all x, y, and z and if C is a circle in the xy-plane, then the circulation of $\vec { F }$ around C is zero.

Use the curl test to determine whether the following vector field is a gradient field. $\vec { F } = y ^ { 3 } z ^ { 2 } \vec { i } + \left( 4 y ^ { 2 } x z ^ { 2 } + 4 y z ^ { 3 } \right) \vec { j } + \left( 2 x y ^ { 3 } z + 3 y ^ { 2 } z ^ { 2 } \right) \vec { k }$

Let $\vec { F } = \left( - 2 x ^ { 2 } - a x z \right) \vec { i } + \left( b x ^ { 2 } + 16 \right) \vec { j } + ( x + c y ) \vec { k }$ be a smooth vector field with $\operatorname { curl } \overrightarrow { \vec { F } } ( 1 , - 4,4 ) = - 3 \vec { i } + 3 \vec { j } + 8 \vec { k }$ Determine the values of a, b and c.

(a)The vector field ${ \vec { F } } = 2 x \vec { i } + 4 y \vec { j } + a z \vec { k }$ has the property that the flux of $\vec {F }$ through any closed surface is 0.What is the value of the constant a? (b)The vector field $\vec { G } = ( a y + b z ) \vec { i } + ( 2 x + 4 z ) \vec { j } + ( 4 x + c y ) \vec { k }$ has the property that the circulation of $\vec { G }$ around any closed curve is 0.What are the values of the constants a, b and c?

Which of the following vector fields has the following properties: 1)the largest value of its circulation density at (1,2,1)is 50 2)the largest value of its circulation density at (1,2,1)occurs around the direction $\frac { 3 } { 5 } \vec { i } + \frac { 4 } { 5 } \vec { j }$

On an exam, students are asked to find the line integral of ${ \vec { F } } = 2 x y \vec { i } + y z \vec { j } + y ^ { 2 } \vec { k }$ over the curve C which is the boundary of the upper hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 2 ^ { 2 } , z \geq 0$ oriented in a counter-clockwise direction when viewed from above.One student wrote: " $\operatorname { curl } { \vec { F } } = - 2 x \vec { j } + 2 \vec { k }$ By Stokes' Theorem $\int _ { C } \vec { F } \cdot \vec{ d r } = \int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A }$ where S is the hemisphere.Since $\operatorname { div } ( - 2 x \vec { j } + 2 \vec { k } ) = 0$ by the Divergence Theorem $\int _ { S } ( - 2 x \vec { j } + 2 \vec { k } ) \cdot \vec { d A } = \int _ { W } 0 d V = 0$ where W is the solid hemisphere.Hence we have $\int _ { C } \vec { F } \cdot \vec{ d r } = 0$ " This answer is wrong.Which part of the student's argument is wrong? Select all that apply.

Let S be the surface of the upper part of the cylinder 4x² + z² = 1, z $\ge$ 0, between the planes y = -1, y = 1, with an upward-pointing normal. (a)Evaluate the flux integral $\int _ { S } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) \cdot \vec { d A }$ (b)Consider W, the solid region described by -1 $\le$ y $\le$ 1, 4x² + z² $\le$ 1, z $\ge$ 0.Evaluate $\int _ { W } \operatorname { div } \left( - 3 x y ^ { 2 } z \vec { i } + 7 \cos z \vec { j } + 3 \vec { k } \right) d V$ Does this contradict the Divergence Theorem? Explain.

Suppose a vector field $\vec { G }$ is always perpendicular to the normal vector at each point of a surface S.What is the value of $\dot { \mathrm { Q } } { \vec { G } } \times \vec { d A } ?$

An oceanographic vessel suspends a paraboloid-shaped net below the ocean at depth of $1200$ feet, held open at the top by a circular metal ring of radius $20$ feet, with bottom $90$ feet below the ring and just touching the ocean floor.Set up coordinates with the origin at the point where the net touches the ocean floor and with z measured upward.

If ${ \vec { F } } = 5 y e ^ { x ^ { 2 } }{\vec { i } } + 4 x y e ^ { y} \vec { j } + 4 z \cos ( x y ) \vec { k }$ find $\operatorname { div } \vec { F} .$

Given $\operatorname { curl } { \vec { F } } = 10 x y z \vec { i } + \left( a y ^ { 2 } z + 2 b y z \right) \vec { j } - 4 z ^ { 2 } \vec { k }$ , find the values of the constants a and b, without knowing the expression of $\vec { F }$ .

Let $\vec { F } = \| \vec { r } \| ^ { p } \vec { r }$ , where p is a positive constant.Is there a value of p such that $\vec {F}$ is a divergence free vector field?

Let $\vec { G }$ be a smooth vector field with $\operatorname { div } \vec { G } = 3$ at every point in space.Find the exponent p in the following: $\frac { \text { The flux of } \vec { G } \text { out of any sphere of radius } r _ { 1 } } { \text { The flux of } \vec { G } \text { out of any sphere of radius } r _ { 2 } } = \left( \frac { r _ { 1 } } { r _ { 2 } } \right) ^ { p }$

You want to build a windmill at the origin that maximizes the circulation of the wind.The wind vector field at any point (x, y, z)in your coordinate world is given by $\vec { F } = ( y + 5 z ) \vec { i } + ( x - 2 z ) \vec { j } + ( 2 y - x ) \vec { k }$ (a)In which direction should you face the windmill to get maximum use from the wind? (b)What will be the strength of the circulation of the wind when you face it in this direction?

State the Divergence Theorem.

Let $\vec { F } = - \frac { x } { \left( x ^ { 2 } + 4 y ^ { 2 } + 3 z ^ { 2 } \right) ^ { 3 / 2 } } \vec { i } - \frac { y } { \left( x ^ { 2 } + 4 y ^ { 2 } + 3 z ^ { 2 } \right) ^ { 3 / 2 } } \vec { j } - \frac { z } { \left( x ^ { 2 } + 4 y ^ { 2 } + 3 z ^ { 2 } \right) ^ { 3 / 2 } } \vec { k }$ Is $\vec {F }$ a divergence free vector field?

Calculate $\operatorname { curl } \overline { \bar { H } }$ where ${ \vec{ H } } = 9 x \vec { i } - 8 x y \vec { j } + x z ^ { 5 } \vec { k }$

Let $\left. { \vec { F } } = \left( y ^ { 2 } + 4 x \cos z \right) \right) \vec { i } + \left( - z ^ { 3 } \right) \vec { j } + \left( z ^ { 2 } - 4 x y \right) \vec { k }$ Find the curl of $\vec { F} .$

Let $\vec { G }$ be a smooth vector field with curl $\vec { G } ( 0,0,0 ) = - 3 \vec { i } - 4 \vec { j } + 4 \vec { k }$ Estimate the circulation around a circle of radius 0.01 in the yz plane, oriented counterclockwise when viewed from the positive x-axis.