# Verifying Divergence Theorem for Given Solid Region and Vector Field?

• Darkstalker86
In summary, you will need to parametrize the side boundary and evaluate the integral for the even functions, and break it into two parts for the different values of z. You can use a similar approach for the bottom boundary. Once you have all the flux integrals, you can verify the Divergence Theorem by comparing them to the result from part A.
Darkstalker86

## Homework Statement

Let E be the solid region defined by $$0 \leq z \leq 9+x^2+y^2$$ and $$x^2+y^2 \leq 16$$.
Let S be the boundary surface of E, with positive (outward) orientation.
Also, consider the vector field $$F(x,y,z)=<x,y,x^4+y^4+z>$$

There are five parts to the problem
A) Compute the $$\int\int\int (div F)dV$$

B) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) upper boundary S1 of E.

C) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) side boundary S2 of E.

D) Compute the flux of $$\int\int F \bullet dS$$of F, across the (oriented) bottom boundary S3 of E.

E) Use A,B,C,D to verify the Divergence Theorem for F on E.

## The Attempt at a Solution

For A)
The Div F = 3. Also, I converted to cylindrical coords, so the limits of integration would be
$$\\0 \leq z \leq 25; \\0 \leq r \leq 4; \\0 \leq \Theta \leq 2\Pi\\$$

$$\int_{0}^{2\Pi}\int_{0}^{4}\int_{0}^{25}(3r) dz dr d\Theta$$
This integral equals 1200Pi.

For B)
$$\iint_{S1} F \bullet dS$$
I parametrized the boundary of the curve as $h(r, \Theta)=<r\cos\Theta, r\sin\Theta,25>$
Then parametrized the vector field with those parameters.
The partials with respect to each variable of h...
$$h^{}_{r}=<cos\Theta, sin\Theta, 0>$$
and $$h^{}_{\Theta}=<-r\sin\Theta, r\cos\Theta,0>$$
Also, I determined the cross product of $h^{}_{r}X h^{}_{\Theta} = <0,0,r>$

So...(this is after parametrizing and dotting with the normal vector)
Question Here: There are factors in this integral that I noticed are odd functions over a symmetric surface. That means that when I integrate them, won't they just go to Zero, and I can save myself time and not do them now correct?
Assuming that is correct, this is the integral I came up with.
$$\int_{0}^{2\Pi}\int_{0}^{4}(25r)*rdrd\Theta$$

I obtained an answer of $$\frac{1600\Pi}{3}$$

Now, I'm not exactly sure that is correct. I know I need to follow the same basic steps in order to determine the side flux and bottom flux integrals. I am having trouble figuring out the flux integral for the side boundary.
Any assistance would be great!

Hello, thank you for your question. I have looked over your work and it seems that you have the right approach for parts A and B. For part C, you are correct in thinking that the integrals for the odd functions will go to zero, but you still need to evaluate the integral for the even functions. For the side boundary, you will need to parametrize the surface as well as the vector field, and then dot them together and integrate over the appropriate limits. I suggest breaking the side boundary into two parts, one for the part where z=0 and one for the part where z=9+x^2+y^2. For part D, you can use a similar approach as part B, but with different limits of integration. Once you have all the flux integrals, you can add them together and compare to the result from part A to verify the Divergence Theorem. I hope this helps.

## 1. 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 over a closed surface in three-dimensional space.

## 2. Why is it important to verify the divergence theorem?

Verifying the divergence theorem is important because it allows us to check the validity of the theorem and ensure that it holds for a given vector field and closed surface. This is crucial in applications such as fluid dynamics, electromagnetism, and heat transfer.

## 3. How do you verify the divergence theorem?

The divergence theorem can be verified by calculating the surface integral of a vector field over a closed surface and comparing it to the volume integral of the field's divergence over the enclosed volume. If the two values are equal, then the theorem is verified.

## 4. What are some common mistakes when verifying the divergence theorem?

Some common mistakes when verifying the divergence theorem include using the wrong orientation for the surface or the wrong direction for the normal vector, misinterpreting the limits of integration, and forgetting to include the appropriate surface or volume elements.

## 5. Can the divergence theorem be applied to all vector fields and closed surfaces?

Yes, the divergence theorem can be applied to any vector field and closed surface, as long as the field is continuous and differentiable over the enclosed volume. However, certain restrictions may apply for special cases, such as when the field is not defined at certain points or when the surface is not a smooth, closed surface.

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