This function fails the conservative test, yet it is path independent?

[flyingpig]

Homework Statement

Show that the line integral is path independent and evaluate the integral

$$\int_C tan(y) dx + x sec^2 (y) dy$$

The Attempt at a Solution

$$\frac{\partial }{\partial x}(tan(y)) = 0 \neq 2xsec^2(y)tan(y) = x\frac{\partial }{\partial y}(sec^2(y))$$

 

[Agent M27]

I believe your problem is stemming from the presence of the differentials, dx and dy. The general form of a line integral is the following:
$$\int_C M dx + N dy$$

Where M & N relate to the quantities in your problem such that:

M = $$tan(y)$$
N = $$x sec^2 (y)$$

Now we know for a force to be conservative it must satisfy the following relationship:

$$\frac{\partial M }{\partial y} = \frac{\partial N }{\partial x}$$

Given this information, is your force conservative?

Hope this helps.

Joe