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Compute the net outward flux of of the vector field $\mathbf{F}=(x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) /\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2} \quad$ across $\quad$ the ellipsoid $9 x^{2}+4 y^{2}+6 z^{2}=36$

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Johns Hopkins University

Missouri State University

Oregon State University

Harvey Mudd College

Lectures

03:04

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x. The input of a function is called the argument and the output is called the value. The set of all permitted inputs is called the domain of the function. Similarly, the set of all permissible outputs is called the codomain. The most common symbols used to represent functions in mathematics are f and g. The set of all possible values of a function is called the image of the function, while the set of all functions from a set "A" to a set "B" is called the set of "B"-valued functions or the function space "B"["A"].

08:32

In mathematics, vector calculus is an important part of differential geometry, together with differential topology and differential geometry. It is also a tool used in many parts of physics. It is a collection of techniques to describe and study the properties of vector fields. It is a broad and deep subject that involves many different mathematical techniques.

11:03

Compute the net outward fl…

06:27

Calculate the net outward …

06:43

01:30

Use the Divergence Theorem…

never before in vector field. Well defined the X. I'm sorry. Over, um, squared off X squared. Plus why, squirt? Plus, she scored the third. Plus why I or the same on the sea. Okay, so pull the vehicle X squared, plus y squared plus c squared Dara to be our half r squared, huh? He's on the bottom Is the same song Are square to death three halfs. Um uh, she would like to You did great in the great flocks The slugs out of the delete. Sorry. Now described by nine X squared. That's four. What? A squared classic C Square Square. A single clue 36 was though these looks like you have here. Thanks. Why? See is like a we come in like, uh, you know, any time you like, make a different mission of home My goal. But so, uh, would like to interpret he's out of Ah, you go. This and the outer normal vector on DDE This region we call it s these flu bugs out off his, uh you can't go in during the boundary of these lips. Sorry. This is yes, is just going like that. They know over that boundary you do, if not credible. Thin. You signal so, uh, by the river. Just fear. And this will be equal to integrating inside of the Arab side with a very generous welfare, which is, uh, Scott, our function you be inside there it was the first we have to find Let's find what? He's the virgins. So, uh, uh, if you'll be reading us X, I over our squirt, put it through half. That's why I where I scored with three halves, we'll see. Okay, over r squared with three halves so that, uh, that raises a legal tool. There are people fags with respect works. Thems are square for this. We have by the course agent who miners six times baby channel yo off three halves R squared three treehouse minus one. And in the times the partial you respect to excel fart squared. So hold that divided by our square To the three halves on that squared. So these for the eggs for the why would be the same part shovel Why were spent the Why which is one I'm sorry squared with three house minus. Why damn so three house of r squared of the free house miners. One she saw half is one. Huh? These number or hear these? Admirable. He's 1/2 1 Huh? Uh, them's rto Have them's ourselves, Inspector y ove r squared Moger r squared three house running that squared. Plus, uh, so one times are square grew three house minus Z terms greenhouse r squared to the 1/2 times are so Mubarak Square with respect to see So first I'll It's, uh I'm forgetting something Nobody but r squared with the three house squared. So first off, let's compute. What are these? These guys here. So why so far are square wheels, but proxies is the partial respect works off x squared. It was White Square Bless he squared, which will be ableto X. Um, I'm not lose the bar shovel of our school respect. Why will be equal to to why Z with this Marcello virus square with respect disease too. So that, uh, putting together all this information You have the double versions these on my virgins. More fair vehicle to, uh yeah. Huh? These number. This is a goto one. He said, here is one. So we'll have three of these times, so no, I didn't Everything so common denominator. So we have three times are square 33 house. I don't know. Well, these, uh, it is not over here will be, Ah, two eggs to Why? To see sort of the stool councils that do. 22 two, two. Sort of the brilliant urging. I got it in terms you will have three times our square The 1/2 uh, times x squared when they were here your time plus y squared plus c squared. I'm all of that divided by our square 3 3/2 from it's going that way so well, uh, that's what about on the next space. So we have the divergence will be cold room. These r r squared to the one house sexes were y squared. Let's see is quote So the virgins physical tow this. Ah, but here will know this at this time here, the people who are squared which is equal Drew r squared today. You were too And so so that we put together the store in terms here. These one I'm dad One we have our square did in one house. Thems are squared two halves. The sequel to R squared to the three house. So that, uh, thes town he's over here will be ableto aren't square. They're three halves on so that ive castles with these terms. Tow. This time that channel, that's toe exam zero or R squared or three halves squared. It is good to see you. So here we see you see that these is a very sensitive girl tells you so that bad injury Elizabeth will be in jail of zero over Be so It's good to see what times the volume this which is you go through equal to Syria. Sort of the these flu bugs zero.

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