[siev]

Imagine a function z=f(x,y) in 3D space. Now picture a plane at say, x=3, that is parallel to the plane passing through the Y and Z axes. This x=3 plane cuts through our function, and its intersection with the z=f(x,y) function forms a sort of 2D function z=g(x)=f(3,y).

(The Wikipedia page[1] has nice images of this [2])

The slope of this new 2D function on the x=3 plane at some point y is then the partial derivative ∂z/∂y for constant x at the point (3,y). As we are "fixing" the value of x to a constant, by only considering the intersection of our original function with a plane at x=x_0.

[1] https://en.wikipedia.org/wiki/Partial_derivative

[2] https://en.wikipedia.org/wiki/File:Partial_func_eg.svg

[slooonz]

That’s just the standard partial derivative in multivariable calculus. This one I have no trouble to understand. My question is about "at constant something" as used in thermodynamics, where "at constant something" is clearly doing more work than just "partial derivative". What work ? How ? Damned if I know.

Consider f(x,y,z), let’s say f(x, y, z) = x^2 + 3y^3 - e^(-z). What’s the difference between "the partial derivative of f with respect to x" and "the partial derivative of f with respect to x at constant y" ? The first one is already at constant y !

In standard multivariate calculus, the partial derivative of f with respect to x , as you explained, is always "at constant y and z".

In thermodynamics, you can say things like "partial derivative of pressure with respect to volume" and add "at constant temperature" or "at constant entropy" and get different results. What ? Why ? How ?

[msravi]

> things like "partial derivative of pressure with respect to volume" and add "at constant temperature"

They're the same thing, isn't it? Except that with add the "at constant temperature" addendum, you're just making explicit the other variable(s) that can potentially be varied. Without it, it just means all other variables, whatever they may be, are constant.

But if something depended on both temperature and some other quantity X, and you said "partial derivative of pressure with respect to volume at constant temperature," that would be sort-of misleading because you're only exlicitly mentioning one of the other two variables - rather, you should say "at constant temperature and X" or not mention either of them.

[bubblyworld]

They aren't the same thing since the first is strictly speaking not well defined - see my answer to the OP. I think the problem is that physicists use the same letter, say U, to denote multiple different mathematical functions depending on the context. The "holding XXX constant" thing serves to tell you which function you're dealing with formally.

[otisv]

I think you mean z=g(y)=f(3,y) instead of g(x)