Exam 20: The Curl and Stokes Theorem

Let $\left. \vec { F } = \left( e ^ { y } + 5 x \right) \right) \vec { i } + \left( 4 y + z \cos x ^ { 5 } \right) \vec { j } + ( 2 z + 5 x ) \vec { k }$ Calculate the flux $\int _ { S } \vec { F } \cdot \vec { d A }$ , where S is the sphere (x-2)² + (y-3)² + z² = $4$ oriented inward.

Let S be the closed surface which is the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ with $- 2 \leq \boldsymbol { z } \leq 1$ topped by the disk in the plane $\bar { z } = 1$ , oriented outward.Then the flux of $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ through S is:

Suppose that the flux of a smooth vector field $\vec { F}$ out of a sphere of radius r centered at the origin is $a r + b r ^ { 2 }$ where a and b are constants.If ${ \vec { F } } = \operatorname { Curl } { \vec { G } }$ for a smooth vector field $\vec { G } ,$ find the values of a and b.

If $\operatorname { cur } 1 { \vec { F} } = \vec { 0 }$ then by Stokes' Theorem the line integral $\dot { \mathrm { Q } } { \bar { F } \times \vec { d r } }$ is equal to zero, where C is the curve y = x², for 0 $\le$ x $\le$ 2.

Let $\vec { F } = 4 z \vec { i } - 4 x \vec { k }$ Let C be the circle of radius a parameterized by x = a cos t, y = 0, z = a sin t, 0 $\le$ t $\le$ 2 $\pi$ and let S be the disk in the xz-plane enclosed by C, oriented in the positive y-direction. (a)Evaluate directly $\int _ { C } \vec { F } \cdot d \vec { r }$ (b)Evaluate directly $\int _ { S } \operatorname { curl } { \vec { F } } \cdot d \vec { A }$ (c)Do these results contradict Stokes' Theorem?

Let S be the boundary surface of a solid region W with outward-pointing normal.Using an appropriate theorem, change the following flux integral into volume integral over W. $\int _ { S } \left( ( 4 x + \cos z ) \vec { i } + \left( 3 y + \sin x ^ { 2 } \right) \vec { j } - 5 z \vec { k } \right) \cdot d \vec { A }$

True or false? If $\operatorname { div } \vec { F } = 4$ for all x, y, z and if S is a surface enclosing a volume V, then $\int _ { S } \vec { F } \cdot \vec { d{ V } } = 4 V$

The figure below shows a vector field of the form

Suppose that $\vec { F }$ is a vector field defined everywhere with constant negative divergence C.Decide if the following statement is true. There is $\text { no oriented }$ surface S for which $\hat { \mathrm { Q } } { \vec { F } \times \vec{ d A } } = 0$

Suppose that $\vec { F }$ is a smooth vector field, defined everywhere. It is possible that $\int _ { S } \vec { F } \cdot d \vec { A } = 3 r ^ { 2 } + 2 r$ , where S is a sphere of radius r centered at the origin.

Let P be a plane through the origin with equation $a x - 4 y + 3 z = 0$ Let $\vec { F }$ be a vector field with curl ${ \vec {F} } = 3 \vec { i } - 5 \vec { j } + 2 \vec { k }$ Suppose $\dot {\mathbf{Q} } {\vec{F} \times \vec{d r}=0}$ for any closed curve on the plane $a x - 4 y + 3 z = 0$ Using Stokes' Theorem, determine the value of a.

(a)Is $\vec { F } = F _ { 1 } ( y , z ) \vec { i } + F _ { 2 } ( x , z ) \vec { j } + F _ { 3 } ( x , y ) \vec { k }$ a divergence free vector field? (b)Do all divergence free vector fields have the form of the vector field in (a)? (c)If $\vec { F }$ has the form given in (a)can we conclude that $\int _ { S } \vec { F } \cdot \vec { d A } = 0$ for any closed surface S?

Consider the two-dimensional fluid flow given by $\vec { F } = \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } x \vec { i } + \left( x ^ { 2 } + y ^ { 2 } \right) ^ { a } y \vec { j }$ where a is a constant. (We allow a to be negative, so $\vec { F }$ may or may not be defined at (0, 0).) (a)Is the fluid flowing away from the origin, toward it, or neither? (b)Calculate the divergence of $\vec { F}$ .Simplify your answer. (c)For what values of a is div $\vec { F}$ positive? Zero? Negative? (d)What does your answer to (c)mean in terms of flow? How does this fit in with your answer to (a)?

Let $\vec { a } = \alpha _ { 1 } \vec { i } + \alpha _ { 2 } \vec { j } + \alpha _ { 3 } \vec { k }$ be a nonzero constant vector and let $\vec { r } = x \vec { i } + y \vec { j } + z \vec { k }$ .Suppose S is the sphere of radius one centered at the origin.There are two (related)reasons why $\int _ { S } ( \vec { a } \times \vec { r } ) \cdot d \vec { A } = 0$ .Select them both.

Suppose that $\vec { F }$ is a vector field defined everywhere with constant negative divergence C.Decide if the following statement is true and explain your answer. $\hat {\mathrm { Q }} { \vec { F } \times \vec{d A} } < 0$ for every $\text { oriented }$ surface S.

Let $\vec { F } = 5 x \vec { i } + z \vec { j } + ( y + 5 x ) \vec { k }$ Use Stokes' Theorem to find $\dot { \mathrm { Q } } { \vec { F } \times \overline { d r } }$ where C is a circle in the xz-plane of radius $6$ , centered at $( 6,0 , - 2 )$ oriented $\text { clockw ise }$ when viewed from the positive y-axis.

Suppose W consists of the interior of two intersecting cylinders of radius 2.One cylinder is centered on the y-axis and extends from y = -5 to y = 5.The other is centered on the x-axis and extends from x = -5 to x = 5.Let S be the entire surface of W except for one circular end of one cylinder, namely the circular end centered at (0,5,0).The boundary of S is therefore the circle $x ^ { 2 } + z ^ { 2 } = 4 , y = 5$ ; the surface S is oriented outward. Let $\vec { F } = \left( 3 x ^ { 2 } + 3 z ^ { 2 } \right) \vec { j } = \operatorname { curl } \left( z ^ { 3 } \vec { i } + y ^ { 3 } \vec { j } - x ^ { 3 } \vec { k } \right)$ . Then $\int _ { S } \vec { F } \cdot d \vec { A } = Q \pi$ .Find Q.

Suppose $\vec { G }$ is a vector field with the property that $\vec { G } = 10 z \vec { k }$ at every point of the surface $\| \vec { r } \| = 0.5$ If $\operatorname { div } \vec { G } = c \text {, }$ where c is a constant, find c.

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 divergence of $\vec { F } .$

Find $\operatorname { curl } \left( - 3 y \vec { i } + 3 x \vec { j } + \left( x ^ { 5 } + y ^ { 5 } + z ^ { 5 } \right) \vec { k } \right)$