How to Determine if a Differential Form is Exact

[zeshkani]

how would i show that this is the exact differential

z=xy-y+lnx+2
dz= (y+ 0)dx + (x-1)dy (i hope the total differential is right)

so how would i show that dz is the exact differential ?

 

[quasar987]

The definition of the total differential of a function f(x,y) is the expression

$$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$

Now with the z you're given, is the expression for dz correct? That is to say, is

$$\frac{\partial z}{\partial x}=(y+0)$$

and

$$\frac{\partial z}{\partial y}=(x-1)$$

??

 

[zeshkani]

i believe dz is correct because i used partial differentials to solve it, and i hope its correct

 

[quasar987]

what is the derivative of ln(x)?

 

[HallsofIvy]

Once you fixed the derivative of ln x thing, the fact that dz is the total derivative of z pretty much means it is an "exact" differential. That's the definition of "exact" differential!

 

[zeshkani]

thx and the derivative of lnx is just 1/x

 

[HallsofIvy]

Yes.
z=xy-y+lnx+2 so
dz= (y+ 1/x)dx + (x-1)dy

And that is an "exact" differential precisely because it is the differential of z.

Now suppose you were given the differential form (y+ 1/x)dx+ (x- 1)dy without having been given z. How would you determine whether it was an exact differential?
If
$$(y+ 1/x)= \frac{\partial z}{\partial x}$$
for some function z and
$$x-1= \frac{\partial z}{\partial y}$$
then
$$\frac{\partial(y+ 1/x)}{\partial y}= \frac{\partial^2 z}{\partial x\partial y}$$
and
$$\frac{\partial(x-1)}{\partial x}= \frac{\partial^2 z}{\partial y\partial x}$$
and so must be equal. Fortunately, it is easy to see that each second derivative is 1 and so they are in fact equal.
Okay, now, how would we find z? Knowing that
$$x-1= \frac{\partial z}{\partial y}$$
integrating with respect to y (treating x as a constant) we get z= xy- y+ C, except that, since we are treating x as a constant, that "constant of integration", C, may depend on x: z= xy- y+ C(x). Differentiating that with respect to x,
$$\frac{\partial z}{\partial x}= y+ C'(x)= y+ 1/x$$
Notice that the "y" terms cancel (it was the "cross condition" above that guarenteed that) and so we have C'(x)= 1/x. Then C(x)= ln(x)+ C where "C" now really is a constant.
Any z(x,y)= xy- y+ ln(x)+ C satisfies dz= (y+ 1/x)dx+ (x-1)dy.