**Igor Kriz**

**Regular problems:**

**1.**

Compute the following integrals:

(a)

(b)

(c) .

**2.**

Using integration of rational functions, compute:

(a)

(b) .

**3.**

Suppose is such that
||*f*(*t*)||=1 (the Euclidean norm ) for every *t*.
Prove that for every .

**4.**

Define for ,
.
Similarly, for ,
, let . (This is called the *Laplacian*.)
Prove that

**Challenge problems:**

**5.**

Integrate:

(a)

(b)

(c) .

**6.**

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);

(c)

Mon Mar 30 11:14:18 EST 1998