Compute the following integrals:
Using integration of rational functions, compute:
Suppose is such that ||f(t)||=1 (the Euclidean norm ) for every t. Prove that for every .
Define for , . Similarly, for , , let . (This is called the Laplacian.) Prove that
Put f(0,0)=0, and
if . Prove that
(a) are continuous in ;
(b) and exist at every point of and are continuous except at (0,0);