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 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.