Expert: Paul Klarreich - 1/21/2007

Question
Hello,

I hope this question is in-category and not too simple, but anyway, here it goes:

Given a vector field F(x,y)=P(x,y)i + Q(x,y)j, where i and j are the unit vectors of the coordinate system, with potential function, I will denote it, G(x,y), such that P(x,y)=DG/Dx, Q(x,y)=DG/Dy (the D's, of course, denoting partial derivatives), for a conservative vector field the potential function can be found by solving the exact DE P(x,y)dx + Q(x,y)dy = 0. The question is, why is it that P(x,y)i + Q(x,y)j is the same as P(x,y)dx + Q(x,y)dy, that is, why does it appear that the unit vectors and the dx - dy signs are the same thing? (Or am I getting something wrong?)

Thanks,

Martin

Answer
Hi, Martin,
Sorry for the delay, but I really wasn't sure what you were asking. (I'm still not sure, but ....)

When you write  P(x,y)i + Q(x,y)j, and you are pretty sure that there is a potential function G(x,y), then this item is the gradient of G, which is a vector derivative.  [It represents the opposite of the way water would flow down the surface.)

P(x,y)dx + Q(x,y)dy  is the total differential and represents the rate of change of height of your G(x,y).  The dx,dy, and the i,j are not the same thing, of course, since the i,j, are vectors (indicating directions) and the dx, dy are just differentials.

Obviously, the two things are related, because they both involve rates of change of the same thing - G(x,y) - and have to be based on the partial derivatives.

I hope this helps somewhat.