$$\oint\_{\partial D} P\;dx + Q\;dy = \iint\_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\;dA$$

Is it correct to interpret this equation as relating the surface area of a three dimensional object (a subset of R3) to the closed path around the object (or the perimeter)? 

Assuming this is the correct interpretation, why does one need to concern themselves with the clockwise/counter-clockwise convention when:

1) The start and endpoint of the path are the same, regardless of how one travels the boundary of the object.
2) What meaning could a negative surface area have?

update: I know wikipedia tells you this a two-dimensional formula, yet from the left side of the equation the integral would look like it resolves into some function F(x2,y2) - F(x1,y1) and furthermore I don't see the use of orientation or vector calculus in 2d when one can use non-vector calculus to find areas in 2d.