Question

Determine whether or not f is a conservative vector field. if it is, find a function f such that f = ∇f. if it is not, enter none. f(x, y) = (yex + sin y) i + (ex + x cos y) j

Answer 1

The given field is
$f(x,y) = (y \ e^{x} + sin(y) \hat{i} + (e^{x}+x \, cosy) \hat{j}$

Note that if the field is of the form
$\vec{f} = P\hat{i} + Q\hat{j}$
then the condition for the field to be conservative is that
$\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$

For the given field,
$\frac{\partial P}{\partial y} =e^{x} + cos y \\ \frac{\partial Q}{\partial x} = e^{x}+cosy$

Because the condition is satisfied, the given field is conservative.
To find the function that satisfies
$f= \bigtriangledown f$

$f(x,y) = \int P(x,y)dx = ye^{x} + x\, siny + g(y)$
Because $\frac{\partial P}{\partial y} = \frac{\partial Q}{\partial x}$, therefore
$\frac{\partial f}{\partial y} =e^{x} + x \, siny + g'(y)$
Because $\frac{\partial f}{\partial y} =Q = e^{x} + x \, cosy$
therefore
g'(y) = 0
g = c  (constant).

Answer:
(a) The given function is conservative.
(b) The function is
$f(x,y) = ye^{x} + x \, siny + c$
where c =  constant.