- #1

Addez123

- 199

- 21

- Homework Statement
- $$A = grad(\frac{1} {\sqrt{(x-3)^2+(y+1)^2+z^2}} + xy^3)$$

Calculate the flow out of a sphere with radius 3, centered at (2, 1, 1)

- Relevant Equations
- Possibly gauss theorem

The issue is that the singularity is not in the center of the sphere.

So how would I calculate it?

I have a few questions:

1. Can I calculate the terms separately like so:

$$A = grad(a+b) = grad(a) + grad(b)$$

2. If I use a spherical coordinate system with the center being at the singularity I can calculate the gradient of first term as

$$grad(a) = -1/r^2 e_r$$

The second term, xy^3, can be calculated with normal coordinates:

$$grad(b) = (y^3, 3xy^2, 0)$$

This creates a few issues though.

I need to convert BOTH those vectors into spherical coordinates with the center being (2, 1, 1).

In grad(b) I recon all I have to do is replace x with ##rsin\theta cos\phi - 2##?

Shouldn't I have to apply scale factors since I'm going from normal coordinates to sphericals?

in grad(a) I just have no idea how to do it.

e_r is not the same since I've moved the center of the spherical system.

So I have no idea how to solve it.

I COULD use spherical coordinates with center (2, 1, 1) and just insert those x, y, z values into A and brute force solve the equation but it really doesn't feel like what I'm suppose to do.

P.S. Sorry for posting so much, I have no class to go to or teacher to ask atm.

So how would I calculate it?

I have a few questions:

1. Can I calculate the terms separately like so:

$$A = grad(a+b) = grad(a) + grad(b)$$

2. If I use a spherical coordinate system with the center being at the singularity I can calculate the gradient of first term as

$$grad(a) = -1/r^2 e_r$$

The second term, xy^3, can be calculated with normal coordinates:

$$grad(b) = (y^3, 3xy^2, 0)$$

This creates a few issues though.

I need to convert BOTH those vectors into spherical coordinates with the center being (2, 1, 1).

In grad(b) I recon all I have to do is replace x with ##rsin\theta cos\phi - 2##?

Shouldn't I have to apply scale factors since I'm going from normal coordinates to sphericals?

in grad(a) I just have no idea how to do it.

e_r is not the same since I've moved the center of the spherical system.

So I have no idea how to solve it.

I COULD use spherical coordinates with center (2, 1, 1) and just insert those x, y, z values into A and brute force solve the equation but it really doesn't feel like what I'm suppose to do.

P.S. Sorry for posting so much, I have no class to go to or teacher to ask atm.