Understanding Divergence and Gradient in Vector Fields

[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?

 

[Defennder]

A divergence is evaluated of a vector field, while the gradient (assuming you mean grad) is done for scalar fields. A related operation, the curl is performed on a vector field.

So we have:
curl: vector field -> vector field
div: vector field -> scalar field
grad: scalar field -> vector field

I'm wondering if there is any defined operation such that we can get a scalar field from a scalar field?

 

[HallsofIvy]

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 $$F=2xi+2yj+2zk$$?

No. the diverence of this vecor field is the scalar function $\nabla\cdot F= 2x+ 2y+ 2z$. The "$\cdot$" in that notation is to remind you of a dot product: the result is a scalar.

And if it is, than what is the gradient?

The gradient is, in effect, the "opposite" of the divergence: it changes a scalar function to a vector field: at each point $\nabla f$ points in the direction of fastest increase and its length is the derivative in that direction.

Notice that if you start with a scalar function, the gradient gives a vector function and you can then apply the divergence to that going back to a scalar function:
[tex]\nabla\cdot (\nabla f)= \nabla^2 f[/itex]
called the "Laplacian" of f. That is a very important operator: it is the simplest second order differential operator that is "invariant under rigid motions".

 

[asi123]

Got it, thanks.