Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across the boundary of the region $D$.Sphere $\quad \mathbf{F}=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k}$$D:$ The solid sphere $x^{2}+y^{2}+z^{2} \leq a^{2}$

Calculus 1 / AB

Calculus 3

Chapter 15

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Integrals

Vectors

Vector Functions

Johns Hopkins University

Harvey Mudd College

Idaho State University

Boston College

Lectures

02:56

In mathematics, a vector (…

03:04

In mathematics, a function…

14:44

Use the Divergence Theorem…

07:12

04:50

10:47

17:47

06:43

In Exercises 9-20, use the…

03:48

In Exercises $5-16,$ use t…

06:24

In Exercises 9- 20 , use t…

11:16

In Exercises $9-20$ , use …

09:56

In Exercises $9-20,$ use t…

09:38

04:55

03:36

05:20

02:49

01:22

00:08

01:30

02:42

02:14

all right, We have our vector function, and we have our region of space that we're trying to find the flux over. So this point, we're, you know, we're gonna be using the diversions. The're, um So let's just find the divergence of about disturb divergence of that, yes, of our first term extent, third musty or D Y number second term, Which is why to the third power, plus our third term, Which three? E over DZ. See? So these air pretty easy driven hours. We just have three x square plus three y squared, plus three. Next, we're gonna evaluate our A region of space. So we have a function of here, So this is gonna be a spear with a radius of A. So we look at this graphically, it'll have the shape. Yeah, kind of like this. And then we put where X and y down men are See, we're gonna end up switching this into spherical coordinates because we can't solve this volume using Cartesian coordinates. So in order to do that, we started our origin. They want a point say, at this part of our cylinder, this is gonna be right here little higher here dropped down is going to go out like that little square going to that point. So this physical coordinates X is gonna be cool too far. Times co sign If I times the sign of data for why we are sine phi to sign, he's just gonna be our science data. Now they are. And this is gonna be the distance to the outer point in our sphere. Or if I is gonna be this angle in between our x axis in the diagonal out toe, our point kind of hard to see in this craft. Just picture it as the point on the X Y plane or the angle in the X Y plane. And then our theta is gonna be the angle downward from the Z axis to our vector in the region of space. So our next goal is gonna be to change. Our are to evaluate our bounds. So first we have our I mean in generics spherical coordinates. It's gonna be from zero to our However, in we have our region bounded of space, which, let me remind you, is X squared plus y squared plus C square just less than or equal to a square. So because that's a function of a sphere with radius A, we know that our radius, it's gonna go zero today, and everything else is just like a normal spear. So if I was going to go from 0 to 2 pi and then data is going to go from zero to pi, and then this is all using her spherical coordinates differential volume, which is our square shine data. We are deify and then defeat a If I can fit it in there. So the last thing to do before we started valuing our flux is we're gonna change our divergence. What we found to be three x squared plus three wise ordered plus threes eastward. We're gonna put that indescribable coordinates. So our divergence, which first all factors. Three hour. We have X squared plus y squared plus C squared in that spherical coordinates. X squared plus y squared plus C squared is just r squared. So our divergence is gonna be reevaluated. 23 r squared. If you have trouble seeing where the X squared plus y squared plus C squared comes from, just go up here and plug in our coordinates. Our new coordinates for X, y and Z and you'll find that you end with three r squared. Come No, we know by the diversions here on the flux is equal to the divergence for a vector field. I'm TV No, that our divergences three r squared times are squared sine of data. Thea, just go through You're 35 de theater so we can see inside our innit roles that we have just kind of multiplication factors of each other. By that, I mean we can factor at the entire integral are which where are bounds air zero a and then if we have take out, all are ours in the three cause means to go somewhere we have three are to the fourth Sometimes they are that if we factor out our theater interval No, it goes from zero to pi We see that we have a sign of data Dictator. Lastly we know we have our co sign We're sorry our fine on our co sign 0 to 2 pi And then there's no fi dependence in our equation So we just have defied So all these integral zehr pretty easy to solve with themselves We have three physically fishing will be factored out. We have the integral of our to the before, which is just are fifth power divided by five, and that's from zero to a next. Integral. We have general science data, which is just a negative sign of failure, and that's from zero to pipe. Then we just have our defy in a role, which turns out we just have to pai minus zero or two point. So next we have three times 1/5 power five now Times negative co sign of pie. Which co center pies, negative ones that just becomes a positive one. Minus are negative Coast and zero, which is just a collective one and then or five, which is just by. And now we get one minus and negative one from the inside term, so that's just too. So we have three times 1/5 over five times, two times two pi. We multiply all those together we get 12 high A to finish all divided by in that is our flux through a sphere that has a readies of a

View More Answers From This Book

Find Another Textbook

In mathematics, a vector (from the Latin word "vehere" meaning &qu…

In mathematics, a function is a relation between a set of inputs and a set o…

09:17

Find the center of mass and the moments of inertia about the coordinate axes…

01:15

Let $C$ be the ellipse in which the plane $2 x+3 y-z=0$ mects the cylinder $…

03:29

Match the differential equations with their slope fields, graphed here. (GRA…

04:29

Find the divergence of the field.The gravitational field in Figure 15.9 …

09:02

(a) find the spherical coordinate limits for the integral that calculates th…

03:57

Use the surface integral in Stokes' Theorem to calculate the flux of th…

05:33

A spring of constant density $\delta$ lies along the helix$$\mathbf{r}(t…

04:37

Set up the iterated integral for evaluating $\iiint_{D} f(r, \theta, z) r d …

04:33

Use the Divergence Theorem to find the outward flux of $\mathbf{F}$ across t…

05:06

sketch the region of integration, reverse the order of integration, and eval…