Why Is It Crucial to Specify Constants When Varying Variables?

[pervect]

We get a lot of questions on PF as to "what happens when you vary foo". I'd like to talk a bit why it's important to specify what you're holding constant when you "vary foo", by referencing elementary partial differential equations.

I realize that this approach might not reach the maximum number of people :-(, but I'm not sure what to do about that to expand the audience. But I'm hoping it will be of some interest anyway. We get our share of posters who don't know what a partial differential equation is all about, but we also get our share of posters who DO know.

The inspiration for this was something I read in Penrose's book, "The Road to Reality" about the second fundamental confusion of calculus (see pg 190).

Suppose we have some function f(x,y), and we want to know how it varies "when we change x". We call this the partial derivative of f with respect to x, and we define it by changing x and holding y constant.

n
If we define X = x, and Y =x+y then

$$\frac{\partial}{\partial x} \neq \frac{\partial}{\partial X}$$

EVEN THOUGH x=X!

If you are used to thinking of partial derivatives as vectors, it becomes really clear, with a simple diagram, why this is so (see attached diagram).

Even if you are NOT used to partial derivatives as vectors, it's hopefully obvious that the vector on the left diagram does represent the concept of "changing x with y held constant", and the other diagram represents the concept of "changing x with Y held constant", and that the two diagrams are NOT THE SAME.So, the bottom line is this Varying x with y held constant is NOT the same as varying x with Y held constant, where Y = x+y. It's different, and if you want to specify what you're varying, you ALSO need to specify what you are not varying, what you are holding constant. At least if you want to get an actual answer.

 

[Drakkith]

Thanks pervect!

 

[Hurkyl]

On the other hand, dx = dX. This is a large part of why I feel that, when thinking algebraically, cotangent vectors are a more fundamental idea than tangent vectors.

 

[Fredrik]

Interesting example. I'm making a mental note of it, in case I need to explain this to someone else.

This sort of thing is the reason why I like to talk about functions and their partial derivatives with respect to the kth variable (and use notations like $D_k\,f$ or $f_{,\,k}\,$) instead of talking about partial derivatives of a variable with respect to some other variable.