# Divergence

• July 27th 2008, 08:51 PM
asi123
Divergence
What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: $F=x^2i+y^2j+z^2k$, the divergence is $F=2xi+2yj+2zk$?

And if it is, than what is the gradient?(Thinking)
• July 27th 2008, 08:57 PM
mr fantastic
Quote:

Originally Posted by asi123
What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk? Mr F says: Yes.

And if it is, than what is the gradient?(Thinking)

You can only take the gradient of a scalar field. You can take the curl of F if you want .....
• July 27th 2008, 09:06 PM
Chris L T521
Quote:

Originally Posted by asi123
What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: $F=x^2i+y^2j+z^2k$, the divergence is $F=2xi+2yj+2zk$?

And if it is, than what is the gradient?(Thinking)

http://img.photobucket.com/albums/v4...e2143a3-21.jpg

Does this clarify things?

--Chris
• July 27th 2008, 09:22 PM
Chris L T521
Quote:

Originally Posted by asi123
What is the Divergence? is it only the Partial derivatives?

Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk? Mr F says: Yes. Chris says: Not quite. This would be your gradient, but not your divergence...
Mr. F, when you take the divergence of a field, you're left with a scalar value...it should be 2x+2y+2z, not 2xi+2yj+2zk...

--Chris
• July 27th 2008, 09:23 PM
asi123
Quote:

Originally Posted by Chris L T521

So one is like the dot product and the other is the cross product?

But still, isnt the gradient already define the partial derivative? why do I need the Divergence?
• July 27th 2008, 09:33 PM
Chris L T521
Quote:

Originally Posted by asi123
So one is like the dot product and the other is the cross product?

This is the proper way of writing it in vector notation. Yes, one is a dot product, and the other is a cross product. Divergence and Curl are important in the study of fluids [this is where the names originated]. Divergence tells you how the fluid flows towards or away from a point; Curl tells you the rotational properties of the fluid. It would make sense that divergence is a scalar, and curl would be a vector.

Quote:

Yes, the gradient consists of partial derivatives, but note the difference between the values of the gradient and divergence:

http://img.photobucket.com/albums/v4...e2143a3-22.jpg

Does this make a little more sense now?

--Chris

EDIT: I should have made this a little clearer by noting that the gradient can only be applied to scalar fields, as Mr. Fantastic had mentioned earlier.
• July 27th 2008, 09:37 PM
asi123
Quote:

Originally Posted by Chris L T521
This is the proper way of writing it in vector notation. Yes, one is a dot product, and the other is a cross product. Divergence and Curl are important in the study of fluids [this is where the names originated]. Divergence tells you how the fluid flows towards or away from a point; Curl tells you the rotational properties of the fluid. It would make sense that divergence is a scalar, and curl would be a vector.

Yes, the gradient consists of partial derivatives, but note the difference between the values of the gradient and divergence:

http://img.photobucket.com/albums/v4...e2143a3-22.jpg

Does this make a little more sense now?

--Chris

Yeah man, that makes a lot of sense.

thanks a lot.
• July 27th 2008, 09:40 PM
mr fantastic
Quote:

Originally Posted by Chris L T521
Mr. F, when you take the divergence of a field, you're left with a scalar value...it should be 2x+2y+2z, not 2xi+2yj+2zk...

--Chris

My mistake. I didn't look closely enough and so saw what I expected to see ......