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a. Show that the outward flux of the position vector field $\mathbf{F}=$ $x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$ through a smooth closed surface $S$ is three times the volume of the region enclosed by the surface.b. Let $n$ be the outward unit normal vector field on $S$. Show that it is not possible for $\mathbf{F}$ to be orthogonal to $\mathbf{n}$ at every point of $S$

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Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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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a. Show that the outward f…

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03:49

05:13

3 (a) Show that the outwar…

03:44

Surface integrals of vecto…

okay, we have our F function, which is equal to X times. I have plus why Times J plus Z k had now the flux of any surface closed over an integral after d n had STS equal to the triple integral volume times that virgins so that everything is just impartial. Max. Expect X. Why would your toe less impartial C respectively, all times TV. So it's a pretty easy personal. Their evidence. We just seem triple in a row, uh, one plus one plus one. You understand what? You just factor that out? Because that's a constant going to get three times the integral where this triple integral is over. Volumes piece. We get three times be any point, any region in space. So and that is our proof. First part, our second part way have to show that it's not possible. Uh, show that is not possible for F to be orthogonal toe end for every point on this. So if f worth diagonal to end hat here, you would get the surface integral zero Yes, which is equal to zero over by the diversions. The're, um, our flux, which is equal to our surface and throw all right. That out by the diversions, dear, this is equal to the virgins of, uh, for every point on s our triple integral. For that, virgins have to be zero. However, R f as we know our sorry are divergence of F, as we know is already equal three, which is three times that volume any point region in space. So in order to have and had an f b orthe article, our volume have to d zero. However, the volume cannot be zero in order to get Siris of clothes surfaces.

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