K. Nicole Clark k-clark

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 **R**^{3}.

(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 **R**^{3}, 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.