a)
is conservative if it is the gradient field for some scalar function
. This would require



Integrating both sides of the first equation with respect to
yields

Differentiate with respect to
:

Differentiate with respect to
:

We want
to be independent of
and
; we can make them both disappear by picking
.
b) This is the so-called triple product, which has the property

Computing the determinant is easy with a cofactor expansion along the first column:

c) Let

Compute the partial derivatives and evaluate them at
:


Then the tangent plane to
at (1, 1, 1) has equation

d) In polar coordinates,
is the set

Then the integral evaluates to

e) By the chain rule,

Eliminating the parameter, we find

so that
when
.
Compute derivatives:





Then at the point (1, 1), the derivative we want is
