Gauss' Law

Gauss’s law for electric field This law is given in two different forms, the integral form and the differential form. Are they identical? I have a feeling that differential form is an approximate version of the integral form. That means, if you solve the differential form you will not get the solution of the integral form. I argue in the following way. While deriving the differential form, the original density function D of the integral form is expanded by Taylor’s series, and then the higher order terms are neglected, assuming that the incremental change goes to zero in the limit. Thus we replace the original non-linear expression for D by a first order linear differential equation. The first order derivatives are decoupled now, and the product terms of the derivatives of D have vanished in the limiting process. Thus the differential form is a linear approximation of the original nonlinear integral form and therefore cannot produce same results. But the books are not saying that. On the contrary books seem to say that they are equivalent forms. Let me know what you think.
- Subhendu Das
West Hills, Ca, USA

The approximation you're describing is the one involved in all calculus problems where you try to express the difference of a function at two points in terms of the derivative at one point in between. However, Gauss' Law is like a 3-D version of the Fundamental Theorem of Calculus, relating the difference at two points to the integral of the derivative over all points in between. Both version of it are exact.

Mike W.

(published on 06/20/2010)