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A function $f(x, y, z)$ is said to be harmonic in a region $D$ in space if it satisfies the Laplace equation$$ \nabla^{2} f=\nabla \cdot \nabla f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0 $$ a. Suppose that $f$ is harmonic throughout a bounded region $D$ enclosed by a smooth surface $S$ and that $\mathbf{n}$ is the chosen unit normal vector on $S$. Show that the integral over $S$ of $\nabla f \cdot \mathbf{n}$ the derivative of $f$ in the direction of $\mathbf{n},$ is zero.b. Show that if $f$ is harmonic on $D$, then$$\iint_{S} f \nabla f \cdot \mathbf{n} d \sigma=\iiint_{D}|\nabla f|^{2} d V$$

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Johns Hopkins University

Campbell University

University of Nottingham

Boston 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.

01:48

A scalar-valued function $…

01:36

A function of two variable…

So you have the phone division, Uh, you say that, uh, function f her morning more d region If, uh, didn't follow me through the civil route. You respect lex theft, remember? Why was the second creative off see of Africa? Respect Dizzy Izzy zero. Ah, but I know that these to greater can be seen us. It ain't virgins. The radiant ff Oh, she's been noted a small black squirt. So we see that harmonic isn't following happens satisfies their luck. Last equation that is called out to us equation. So these side is called couple of another man's aggression. So last equation equals you. So you have the following. If, um, the it is a bounded. Yeah. So what service, huh? Service? Yes. Then, uh, befalling happens. Degrading the greedy int in the unit. Nor more to s. So these were being the floods or the service out of the service. Cigna. Well, this is, uh this is equal to this year because, um bye. Divergence theorem. These were people toe giving over your homeboy the orgy, The big emergence off ground if, um better these diversions off, Brad. Easy goto they the blast crater. You should be zero. Sear them. So the volume or we'll be So it is his, uh, his violin. Better this. I'll be some numbers. It will be some Cranston. I'm zero. Which is the vault is here. So what are these doing? That brilliant faith out off the normal person out off this surface? Easy will go here on, uh, also you have be if, uh if you some are more nick morning function on Do come to me region. Yeah, well, uh, small surface, the service house of business. Uh, before what happens? Uh, then we have that are doing. We're this surface. Yeah, Know them soon, Grete. And two, this is the defense of surface is full doing, or the whole won't be ordered holding me. No, he's a squared. The right over the grabbed off the square. Also, these time here can be seen us in that pronto. If we took the break up, just know that No. So Well, what we gonna do here is to treat, decide, received. If perhaps we can. You can get it outside. So well, you have that. He's ah, job, Victor. So that victor that are within normal inside off. Oh, out off the surface over to surface. Bye. Diversions. Purim vehicle too. Review the virgins of that later. Well, the drug samples Yeah, multiply did really? So the equal toe doing that divergence over plan B and you'll be, but by the identities, identities or farces for the blind. Ah, uh, if a better view we have that piece. Did they bury himself? Some function that's a vector Fuel the origins of these people, though. Well, I have times on today rages well, first, plus, uh, granted Beth about product we, uh, or rectal feel so that, uh, well, we have these years. So you want to compute to die virgins? The virgin's off half right f is grad FSO selector. These three place of that serve these real people toe. You know, Jewel, we're be off f terms they virgins over the great interfere Plus, uh, no. The rating about bird with r f r f. He's great, in fact. So that's better. Devi will be able to I know that this part is already what you want these parties, huh? The greedy it off squared. So this part Morgan Day Savarin Angel Ortiz This is what we want and also note that these in every sense of grad fr that is precisely the definition off the separator. It's a very rare here is equal to the Lavery himself. But ah, we know that our kids are mine. Our money over f So being our money means that, uh, divergence. Oh, God, Yeah, you see Well, d'oh, do you so that these part of people do CEO of themself where volume well, for any continues functions here themselves were about invisible to zero. This time he's going so that these, well, people toe God. And that is that is precisely what I want that these term coincides with into a year over the morning. The oh, they know squared off Brilliant, yeah.

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