My textbook has the following problem:

They apparently mean in . Recall that a "weak solution" (in this case) is a solution which satisfiesShow that for a continuous function the expression is a weak solution of the partial differential equation

[Hint: Transform for the integral

to the coordinates . Use .]

(1)

for every test function , i.e. for every continuously differentiable function with compact support in . However it is (supposedly) sufficient to show that (1) holds for all , i.e. all smooth functions with compact support in .

My question is this: Why can we assume that a smooth (or continuously differentiable) function of two variables can be written as the product of functions of a single variable ? Or am I missing something here ?

Thanks !