What is the Divergence? is it only the Partial derivatives?
Lets say I have a vector field: $\displaystyle F=x^2i+y^2j+z^2k$, the divergence is $\displaystyle F=2xi+2yj+2zk$?
And if it is, than what is the gradient?
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...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...
--Chris
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:But still, isnt the gradient already define the partial derivative?
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.