physical applycation of gradient,divergence,&curl?

Answer 1

Gradient
Imagine you have a latex fabric that is uniformly stretched in a square frame. Mark the fabric with horizontal and vertical lines forming squares. Let the skin be a function and horizontal and vertical sides represent x and y-axis respectively. We can write f(x , y).
If we take the partial derivative and add these partial derivatives we will get

d(f(x , y)/dx + d(f(x , y)/dy = 0 Why is it zero?

The slope everywhere as far as this function is concerned is zero.

If we introduce a pull on a surface the function will no longer have a derivative zero at this point with the slope converging to zero further from the pull point. It is analogous to a room with a candle as a source of heat. The closer to the candle the sharper will be the slope.

Curl
http://en.wikipedia.org/wiki/Curl
Divergence
http://en.wikipedia.org/wiki/Divergence

Answer 2

They are the primary mathematical tools you use to figure out how electric and magnetic fields relate to each other and to matter of various kinds. I spent a full year doing this as part of my undergraduate program in physics and electrical engineering.