Mathematical Physics Vol 1

Chapter 4. Field theory

174

As we can see, the result is the same as under a).

Exercise 115

Prove that

I C

M d x + N d y = 0 ,

for any closed curve C in a simply connected region, iff ∂ N ∂ x = ∂ M ∂ y

is valid in the entire

region.

Proof Assume that M and N are continuous functions with continuous partial derivatives in the entire region R , bounded by the curve C . Then, according to Stokes’ theorem I C M d x + N d y = x R ( ∂ N ∂ x − ∂ M ∂ y ) d x d y .

∂ M ∂ y

∂ N ∂ x

If

in the region R , then

=

I C I C

M d x + N d y = 0 .

Conversely, let

M d x + N d y = 0 ,

for each curve C . Assume that there is at least one point P in R forwhich ∂ N ∂ x − ∂ M ∂ y̸ = 0 .

Then this expression is different from zero also in some neighborhood A of point P , due to the continuity of the functions M and N . If the curve Γ is the boundary of region A then 0̸ = I Γ M d x + N d y = x A ∂ N ∂ x − ∂ M ∂ y d x d y , which is contrary to the assumption that the line integral is equal to zero for each closed curve. It thus follows that ∂ N ∂ x − ∂ M ∂ y = 0 in all points of region R .