Quick Question

1. Jan 30, 2008

Physics_wiz

Say I have a function $$g = g(x,y)$$. Now, I have another function defined as:
$$f(x,y) = \partial / \partial x [g(x,y)]$$. Is the following true:

$$f(a,b) = (\partial / \partial x [g(x,y)])_{x = a, y = b} = \partial / \partial a [g(a,b)]$$

a, b, x, y are all variables. Is this true in general for any function of any number of variables?

2. Jan 30, 2008

dhris

It looks like you just substituted a for x and b for y. So I guess the answer is yes, you can do that :).

3. Jan 30, 2008

neutrino

To me that looks like the partial derivative of g with respect to x, at the point (a,b). It's better without that middle part, if you just want to give the variables a different set of 'labels'.

4. Jan 30, 2008

Physics_wiz

Yes, you're right in your interpretation. However, the equality between the middle part and the last part is the most crucial step for my purposes, so I can't take it out.

5. Jan 31, 2008

HallsofIvy

The only difference between the "middle" and "last" parts of you statement are that in the middle part, you differentiate using the "symbols" x and y, then replace them by a and b, while, in the last part, you replace x and y by a and b, then differentiate. Yes, those are equal.