It's Obvious. Not!

This starts a series of posts that hopefully will add detail if not clarity to a number of physics and math topics. As I have been reviewing several of my physics and math texts, I’ve noticed that there are a number of ‘missing steps’ in some of the explanations and examples. In some cases, they seem to have been overlooked, and in others, the author of the book has actually made that statement dreaded by students everywhere, ‘the derivation is trivial/obvious…’ Each of these posts will take one of these ‘incomplete’ texts and elucidate the, (usually brief), steps that I found useful in understanding them. If you have other clarifications that are helpful, please add them!

This first post relates to an excellent summary of vector calculus titled: “div grand curl and all that: an informal text on vector calculus”. In the second edition of the book on p. 25 the author sets out to take the partial derivatives of:

He immediately arrives at the conclusion that:

My confusion:

x over z? y over z? Where did z come from when taking a partial differential of an equation that contained only x and y?

Here’s what’s missing:

Using the chain rule of differentiation:

and simplifying

and moving the negative power below the denominator:

So far, so good, but where does the z come from?

Using the original problem above:

As a substitution:

The same steps can be followed to arrive at the partial derivative with respect to y.