I want know about the gradient of a scalar field having non parallel 2D contours.?

Answer 1

I'm not sure exactly what you mean or exactly what you want to know, but I'll take a stab at answering you.

Since you mention 2D contours, I'm going to assume that this is a 2D field.

If the field is continuous, then as the normal (perpendicular) distance between the contours approaches zero (i.e. they get closer and closer together), they will become closer to being parallel and the gradient is defined (and computed) in the usual way. That's pretty obvious.

The less obvious case (and associated question) is: What happens to the gradient where there is a discontinuity in the field? Let's suppose the discontinuity is along a straight line. In such a case the contours close to the discontinuity on one side will not (in general) be parallel to those on the other side even as the distance between them approaches zero. The discontinuity in the field corresponds to a singularity in the gradient. It's not defined on the line. Sometimes people say that it's "infinite" along the line.

The limiting value of the gradient as the line is approached from either side is well defined, but there are two (potentially) different values - one for each side of the line.

Is that what you want to know?