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Green's second formula (Continuation of Exercise 29.) Interchange $f$ and $g$ in Equation (10) to obtain a similar formula. Then subtract this formula from Equation (10) to show that$$\iint_{S}(f \nabla g-g \nabla f) \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g-g \nabla^{2} f\right) d V$$This equation is Green's second formula.

Calculus 3

Chapter 16

Integrals and Vector Fields

Section 8

The Divergence Theorem and a Unified Theory

Vector Functions

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Harvey Mudd College

University of Michigan - Ann Arbor

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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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Green's second formul…

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07:55

Prove Green's identit…

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Green's first formula…

So we previously found at the surface integral of f times The grating or G dot and had es is equal by the divergence theorem to the triple volume integral of F tens Little plastic energy plus the grading of G daughters into the ingredient of F. So now we're gonna use that That prove that uh uh, she minus g great. And then this whole quantity dotted into an hat, Yes. Is equal triple volume a role, uh, f plus e g Dynasty posse. Now, with this initial equation from previous questions at the top here, this is pretty easy. So knowing her if we know our rules about vector calculus, we know that we can split up these operations using the properties of vector addition. So we have the surface of our first radiant. And then if we got that in tow hat Yes. And then plus are other part, which is the service in a role of negative que radiant and and hat ds. Now we know by our divergence theorem that this is equal. It's actually over The volume of our divergence of F which I looked back up at the top here can be simplified down to this term. And so we have Beth plastic G plus the grating left daughters into the great energy, do you mean? And now we can use that same property for our second integral and just flip the G in the F so even minus side. And so we factor that out. We get the triple integral of the same volume of Jean Classy in plus ingredient of F dotted into the Grady in key baby. Now, because these have the same bounds, they can be added back together. We have just in the tire equations equals volume integral in Khowst disease, I have radiant squared of G with that in another privacy, plus the greatest dotted into the green of G minus. Next one, which is G. I lost, you know, plus that the grading of F dotted into ingredient G. Now that is all multiplied by our differential volume DP They can't get that already right below. So now we can combine these two. So we have I noticed that we have one positive creating a f dot into greeting of G. And then we distribute this negative sign into here. We have one negative Grady and 1/2 times great unity. So those to actually cancel with each other and left with the volume in a girl? Uh huh. Applause C and a G minus g classy in or the equation that we were trying to remember.

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