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Cylindrical can $\quad \mathbf{F}=\left(6 x^{2}+2 x y\right) \mathbf{i}+\left(2 y+x^{2} z\right) \mathbf{j}+4 x^{2} y^{3} \mathbf{k}$$D:$ The region cut from the first octant by the cylinder $x^{2}+y^{2}=4$ and the plane $z=3$

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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

Campbell University

Idaho State University

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.

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So we have this, William, of, uh, we'll have these. We will find the region. The volume off these region Three in three dimensional space. We have our ceiling there. Where you still very nice. Not really. Seeing there is described right equation that, um, x square plus why square it is more than square. That is four square. Yes, we want to. Could that plane? But he's, uh, like this plane so that this plane is described by, um Why, you see, people go for So these meats at that point there on, uh, 00 or you have there. So for why, for four c, that's for I've been here that point intersection there is zero for do so. Uh, well, let five. It is everything that plain on the plain. Seiko zero. So we're gonna have a portion off that ceiling there. The's bottom on the job, Something like, uh like that. You want to find the volume? Dana, what's convenient to use cylindrical 1/4 of its because here, um, this is really conditions that are square is more to squared. I mean, these would be saying that Go. Why is he go to our sign of better on the excessive Aled our course so that this condition would be saying, Ah, our sign there are last See Mr Equal before, so you can see there. See, uh, moving that there she would be between zero. Because the talk is the bottom is that point this point will be even by son four minus are we'll sign san off so that for the volume, this volume would be, ah, illegal to the angel. So, uh, for C, have a diabetes year on four. Well, it's our sign. That is We are You see, saris, the volume element and then for our well, our since he's positive will go from zero up to because he thinks world on them. So, uh, I would go from syrup to to Rhonda for yourself free throw all the way around. So theater would go from, uh, cereal to buy. So you have these seem trivial. Oh, well, then he had a great first. We see the rest is constant with respect disease. We're gonna thing see so that I've already between zero when four. When A sorrow sign with equal toe. Well, these Well, I'm saying, Does your so we'll be able to Our four minus are sign better make zero. So would be going through that the line integral is gone. And then you have or my notes are site of fairer. You didn't from Syria off to off that times. These are times are they are, um, So you would have these Let me have been trouble. There are. Ah, the smell of these will be all the internal for our so our square scientific data. So the through all these is gonna be the adrenal four R's toe r squared. Because if you differentiating down so you get that on, then intel off. Minus our stories. Miners are cube thirds by the power rule. She would have these evaluated within sirrah, until give me up into little deep from zero to buy. You have these on that, too, is gonna be able to Teoh them's the squared minus to que thirds show, you know, dinner. Then let's hear it off. Minus two with that zero. My serial them sign off there so you would have zero for this part and then we're left with in droves. Was the bodies 44 times. So that read it off age. This is also ate. Thirds might say, thirds saying Fida said in jail of these era from one of the drill of 80 would be just yeah, fitter when they drill over minus sign only 12 of minus. And he's, uh, co sign. So we have Plus goes we can check because if you differentiate these time with respectful data, you get that because, of course, I M is mine assigned so good and then we'll be that evaluated. It would be that some evil we could meeting to buy zero ah, to buy zero that is gonna be able to. So hey, time, sir, to buy. Plus 1/3 course of by that is worm. My mom gave minus sear them saved. Plus did thirds was a serial. So, uh, this is one was a zero. It's one that it's also one cause of what is one. So you would have that day bullion. Where you going to of the 16 by 16 by glossy thirds minus outlaw like that X minus 8/3. Very nice of these rows away. Now that the drill would be photo to stand by. Nobody's, uh well, that is dividing before Holomisa. Well, you 16 by there

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