Understanding the ODE Theorem and Partial Derivatives in Math

[flyingpig]

Homework Statement

[PLAIN]http://img64.imageshack.us/img64/6967/unledyac.jpg

2. The headache

I know that f(x,y) is just any function, but my brain completely collapsed when they introduced $$\frac{\partial f }{\partial y}$$

What does that mean? Why only $$\frac{\partial f }{\partial y}$$? What about $$\frac{\partial f }{\partial x}$$

My book does not do the jsutice of explaining it properly. Also what if I take z = f(x,y)??

 

[HallsofIvy]

Probably your textbook assumes that you have taken Calculus and it does not need to explain what a partial derivative is.

The reason it does not mention $\partial f/\partial x$ is because it is the dependence on the dependent variable that is important. In order to understand why, you would have to look at the proof of the theorem itself.

Actually, the continuity of $\partial f/\partial y$ is a sufficient condition but not necessary. A more precise condition is that the function, f, be "Lipschitz" in y. That is, that there exist a constant, C, such that $|f(x, y_1)- f(x, y_2)|\le C|y_1- y_2|$, for any x, $y_1$, and $y_2$ in some neighborhood of $(x_0, y_0)$. Most elementary textbooks require that the derivative be continuous which then implies Lipschitz.

 

[flyingpig]

No I do know what partial derivatives are, but i just don't understand where this theorem came from .

The delta thing reminds me of the epilson delta limit