One of the above identities says ∇⋅
In verifying this you need to use the following manipulation which will generally hold under reasonable conditions but which has not been carefully shown yet.
The above formula seems plausible because the integral is a sort of a sum and the derivative of a sum is the sum of the derivatives. However, this sort of sloppy reasoning will get you into all sorts of trouble. The formula involves the interchange of two limit operations, the integral and the limit of a difference quotient. Such an interchange can only be accomplished through a theorem. The following gives the necessary result.
Proof: Let Δx be such that x,x + Δx are both in
The second formula of Theorem 27.1.3 states ∇×∇ϕ = 0. This suggests the following question: Suppose ∇× f = 0, does it follow there exists ϕ, a scalar field such that ∇ϕ = f? The answer to this is often yes and a theorem will be given and proved after the presentation of Stoke’s theorem. This scalar field ϕ, is called a scalar potential for f.