# Math Help - Holding variables constant or not

1. ## Holding variables constant or not

In the proof for Snell's law in physics, I don't understand why some variables are held constant and some not. Here's the proof:
Snell's Law - ProofWiki

They're taking the derivative of $T$ with respect to the length $x$, as labeled in the picture above. Why in particular the length $x$ and not e.g. the length $c-x$ or perhaps $a$? How come all the other lengths are considered parameters i.e. why are we holding them in particular constant not $x$ in this context? Why do we choose and consider $x$ to be the actual variabel?

Note: I'm doing a similar assignment where I'm to maximize the area of a triangle and I'm having a bit of a hard time deciding with respect to which variabel I'm supposed to differentiate.

2. ## Re: Holding variables constant or not

I read through that and can find no mention of "x being held constant". In fact, because they take the derivative with respect to x, x can't be constant.

3. ## Re: Holding variables constant or not

Originally Posted by HallsofIvy
I read through that and can find no mention of "x being held constant". In fact, because they take the derivative with respect to x, x can't be constant.
That's not even what I claimed. I wonder why x is considered to be the "main" variabel whereas a, b and etcetera are held constant, despite the fact they in fact are not fixed!

4. ## Re: Holding variables constant or not

Hopefully I can be of some assistance. I understand the problem as follows:

1) A ray of light is emitted from A and is observed at B and there is no confusion about this. In other words, the points A and B are fixed: we know the light leaves A and is seen by B.

2) What we don't know is the path that the light takes from A to B. Luckily we have Fermat's Principle to help us with this!

3) What I think is confusing is that the artist of the diagram is not making it entirely clear what line represents the interface between the mediums - does he/she mean the solid line is the interface or the dashed line? I have concluded that they intend for the solid line to be the interface between the two because $v_{1}$ sits above the solid line and $v_{2}$ below it.

Since the solid line represents the interface, the "variable" a is constant because (using 1 above) A is fixed. Since A is fixed, its height above the solid line (represented by a) should not change. The "variable" b is constant for similar reasons.

From 2 above we recall that we don't know where the light wants to cross the interface. In other words we don't know how far to the right of A it crosses - the artist of the diagram has let this variable be represented by $x.$ This is legitimate variable in the sense that we don't know what value for $x$ will minimize the time. Using Fermat's Principle we know that wherever it crosses will minimize the time. Everything else follows pretty much as stated in the link you posted.

So, I think the most important thing to realize is that since A and B are fixed, the variables $a$ and $b$ are constants, so we would not differentiate with respect to them.

Does this help answer your question? Let me know is anything is unclear.

Good luck!