# Divergence theorem for vector functions

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Surface S and 3D space E both satisfy divergence theorem conditions.

Function f is scalar with continuous partials.

I must prove

Double integral of f DS in normal direction = triple integral gradient f times dV

Surface S is not defined by a picture nor with an equation.

Help me. I don't know how to start this unique integral.

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pasmith
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Consider the vector field $\mathbf{c}f$ where $\mathbf{c}$ is an arbitrary constant vector.

Yes. I tried it. I don't get anywhere after considering vector c is a constant vector.

Orodruin
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Yes. I tried it. I don't get anywhere after considering vector c is a constant vector.
Pull it inside of the derivatives and integrals and use the divergence theorem.

Divergence of constant vector is obviously zero.

The surface integral is ambiguous due to no picture or function that defines the surface.

Orodruin
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Divergence of constant vector is obviously zero.

The surface integral is ambiguous due to no picture or function that defines the surface.
You do not need a picture. It is the boundary surface of the volume that the volume integral is over.

How can the infinite sum of scalar times the surface areas equal the infinite sum of scalar times their volumes?

Also,

Closed double integral of f times constant vector equation C dot dS in normal direction

=

Triple integral of divergence of f times constant vector C dV

Produces triple integral of gradient f dot constant vector C dV on the right side only.

I can't seem to do the surface integral.

Orodruin
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How can the infinite sum of scalar times the surface areas equal the infinite sum of scalar times their volumes?
It is not what you have, you have
$$\oint_{\partial V} f\, d\vec S = \int_V \nabla f \, dV.$$
This is the integral of ##f## over the surface that is equal to the integral of the gradient of ##f## over the volume.

I can't seem to do the surface integral.
You don't need to. You just need to show that they are equal.

Triple integral of divergence of f times constant vector C dV
Please use the LaTeX feature. It is very difficult to read mathematical formulae in text.

Closed double integral of f times constant vector equation C dot dS in normal direction
Produces triple integral of gradient f dot constant vector C dV on the right side only.
Yes, so move the constant vector out of the integrals.

It is not what you have, you have
$$\oint_{\partial V} f\, d\vec S = \int_V \nabla f \, dV.$$
This is the integral of ##f## over the surface that is equal to the integral of the gradient of ##f## over the volume.

You don't need to. You just need to show that they are equal.

Please use the LaTeX feature. It is very difficult to read mathematical formulae in text.

Yes, so move the constant vector out of the integrals.
Ok. So how can the infinite sum of scalar times surface areas equal the infinite sum of gradient of the scalar times the volumes???

Orodruin
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Ok. So how can the infinite sum of scalar times surface areas equal the infinite sum of gradient of the scalar times the volumes???
This is what you are supposed to show!

Do you have the same problem with the regular divergence theorem?

It is really no stranger than an integral of a function being equal to the difference between the values of the primitive functions at the end-points.

I understand the divergence theorem quite well. From 2 closed line integrals to the closed surface integral of the curl of the function to the triple integral of the divergence of the curl of that original function while putting emphasis on zero... I understand divergence theorem very well when I can rid the vectors from the dot product and the divergence Operator.

Here is my problem.

Let's start off with vector function equals scalar f times constant vector.

You told me to put this vector function into the divergence theorem.

Is this correct???

Orodruin
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Let's start off with vector function equals scalar f times constant vector.

You told me to put this vector function into the divergence theorem.

Is this correct???
Please start using the math markup available, see LaTeX Primer. There really is no way for me to tell whether you have understood my meaning or not when you use English instead of math.

This is what you are supposed to show!

Do you have the same problem with the regular divergence theorem?

It is really no stranger than an integral of a function being equal to the difference between the values of the primitive functions at the end-points.
The divergence theorem states that the flux of the vector field through the surface is equal to the divergence of the vector field throughout the volume. So, no I do not have the same problem with the divergence theorem

Orodruin
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The divergence theorem states that the flux of the vector field through the surface is equal to the divergence of the vector field throughout the volume. So, no I do not have the same problem with the divergence theorem
And this one says that a scalar quantity integrated with the surface element is equal to the volume integral of the gradient. It is basically Archimedes' principle. Also, it follows directly from the divergence theorem if you go through the steps that have been outlined above.

If we don't place Archimedes principle on this equality... there really is no meaning.

Orodruin
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If we don't place Archimedes principle on this equality... there really is no meaning.
What do you mean "no meaning"? It is a mathematical identity. What you do with it and what it can be used to model is a different thing - but it definitely has mathematical meaning.

The surface integral

Scalar function times vector dS

Does NOT make sense.

Furthermore, Volume integral of gradient of the scalar function times dV makes no sense either.

Equating these two integrals to each other just does not produce meaning as there is clear meaning of the divergence theorem.

"Total Flux of vector field through the total closed surface is equal to the divergence of the vector field times the volume of enclosed Space"

How do I derive this vector integral from the simple divergence theorem????

I seem to lose the vector if I start off with

Scalar function times arbitrary constant Vector V as my starting vector field.

Orodruin
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Scalar function times vector dS

Does NOT make sense.
This is just wrong. It is a mathematical statement. Whether it makes physical sense to describe something is a different matter. I have already provided you with cases where integrals of this type does make physical sense.
Furthermore, Volume integral of gradient of the scalar function times dV makes no sense either.
Of course it does, it is just an integral of a vector field - the result is a vector. You cannot just blatantly proclaim that it does not make sense.