# How Do You Apply the Divergence Theorem to a Vector Field in a Unit Cube?

• N8G
In summary, the conversation discusses the application of the divergence theorem using a function, v, and a unit cube at the origin. The homework equations state that the flux of vector v at the boundary of the closed surface is equal to the volume integral of the divergence of the vector field. The attempt at a solution involves completing the volume integral and setting up the necessary surface integrals for each face of the cube. The process involves picking a face, expressing the outward normal in Cartesian coordinates, calculating the dot product, and then doing the integral. The conversation also includes a helpful walkthrough for setting up one of the surface integrals.
N8G

## Homework Statement

Griffiths Introduction to Electrodynamics 4th Edition
Example 1.10

Check the divergence theorem using the function:
v = y^2 (i) + (2xy + z^2) (j) + (2yz) (k)
and a unit cube at the origin.

## Homework Equations

(closed)∫v⋅da = ∫∇⋅vdV
The flux of vector v at the boundary of the closed surface (surface integrals) is equal to the volume integral of the divergence of the vector field.

## The Attempt at a Solution

I have completed the volume integral of ∇⋅v and it equals 2. However the issue I am running into is that I do not understand how to properly construct all 6 of the surface integrals necessary to determine the left hand side of the equation. Obviously it must equal 2 but I need help setting up the integrals.

This is an example problem in the Griffiths E&M text however it just shows the integrals and their solutions rather than how each integral is conceived for each surface of the cube.

If anyone can give me a walkthrough on how to set up those integrals, I would appreciate it immensely.

Thank you.

You have to
1. Pick a cube face, say the one parallel to the yz-plane at x = a (a = 1 m).
2. Express the outward normal in Cartesian coordinates. Answer: ##\hat n = \hat x##.
3. Calculate ##\vec v \cdot \vec n##. Answer: ##(y^2 \hat x +(2ay + z^2) \hat y +2yz \hat z)\cdot \hat x##
4. Multiply out the dot product.
5. Do the integral ##\int{\vec v \cdot \vec n~dA},## where for this particular choice of face ##dA=dy~dz##.

Note that in step 3 I replaced ##x## with its constant value at that face. Similar replacements must be done at the other 5 faces where ##x##, ##y## or ##z## are constant.

I'll give you an example of one surface integral: the surface that has ## z=1 ## with the outward normal pointing in the ## +\hat{k} ## direction, where ## x ## and ## y ## each get integrated from ## 0 ## to ## 1 ##. You just take ## \int\limits_{0}^{1} \int\limits_{0}^{1} \vec{v} \cdot \hat{k} \, dx dy ##. It takes a little work doing all 6 faces, and you need to make sure you pick the outward pointing normal in the right direction, but it's pretty straightforward.

## What is the Divergence Theorem?

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field to the volume integral of its divergence.

## How is the Divergence Theorem used in E&M?

In E&M (electromagnetism), the Divergence Theorem is used to relate the electric flux through a closed surface to the charge enclosed within that surface. This is useful in solving problems involving electric fields and charges.

## What is the mathematical formula for the Divergence Theorem?

The mathematical formula for the Divergence Theorem is ∫∫S F · dA = ∫∫∫V ∇ · F dV, where F is a vector field, S is a closed surface, and V is the volume enclosed by the surface.

## What are the applications of the Divergence Theorem?

The Divergence Theorem has many applications in physics and engineering, such as in fluid dynamics, electromagnetism, and heat transfer. It is also used in mathematical proofs and in the development of other theorems in vector calculus.

## Can you give an example of how the Divergence Theorem is used in real-world problems?

One example of the Divergence Theorem being used in real-world problems is in calculating the flow of a fluid through a pipe. By applying the theorem, the flow rate can be related to the divergence of the velocity field, making it easier to solve for the flow rate and analyze the behavior of the fluid.

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