K. Nicole Clark                                                                                             k-clark

4/21/2004

URL: people.tamu.edu/~knc8928/7.5.2.pdf

7.5.2 – The point of this problem is to compile a “cookbook” summary of the discussion of “holes in space”.  Suppose that  is a vector field defined in a region  in R3.

(a) State a list of conditions on  and  that together guarantee existence of a             scalar function f such that  everywhere in.

(b) State a list of conditions on  and  that together guarantee existence of a             vector field  such that  everywhere in .

Solution

(a) 1. Obviously, for such a scalar to exist the vector field  must be defined and smooth             everywhere on R3, and the curl and divergence conditions must be satisfied everywhere as well.

2.  (which is equivalent to saying that the Jacobian matrix is symmetric); this must be true due to the fact that the curl of a gradient is always zero.

3. From Stokes’ Theorem we know that the values of all surfaces integrated over the closed surface  must be the same.

(b) 1. Condition (1) from part (a) is also a condition in this case.

2. ; this is a condition because the divergence of a curl always equals zero.

3. Condition (3) from part (a) is also a condition in this case.