# Mathematics and notation

Despite mathematicians’ apparent focus on notation, I’ve always found it difficult to navigate certain notation in math. In fact, although I am only up to the level of undergraduate mathematics in the US, I might say that the chief difficulty so far has been with notation, not some intrinsic difficulty of the subject (for, if mathematics is merely verbal and a sequence of tautologies, then the ability to follow instructions is all that is necessary—until the instructions are obscured by poor notation!).

Part of the problem is that there is a lot in notation that is *implicit*. When talking about mathematics in person, there is no need to specify all the context, because if anything is unclear, one can just ask and it will be clarified verbally.

I like what James Morrow (my math 334 instructor) said when discussing the multivariable chain rule:

\[ \frac{\partial z}{\partial x_1} = \frac{\partial z}{\partial y_1} \frac{\partial y_1}{\partial x_1} + \cdots + \frac{\partial z}{\partial y_n} \frac{\partial y_n}{\partial x_1} \]

Writing it down this way is okay, but you have to understand what the symbols mean […] It’s an efficient way to write it down […] I think you’ll get used to it […] The exercises [in the book] are meant to show you that it’s easy to mess it up […] Chemists don’t tell you which variable they’re keeping constant […] There’s a lot that’s implicit and it’s hard to tell what’s going on.

I also like AbstractMath.org.

Another example with partial derivatives (inspired by an ambiguity that Terence Tao touched on in his *Analysis II*): define \(f : \mathbf{R}^2 \to \mathbf{R}\) as \((x,y) \mapsto x^2\) and \(g:\mathbf{R}^2 \to \mathbf{R}\) as \((x,y)\mapsto y^2\). Observe that we have \(f(x,x) = g(x,x) = x^2\). Trick question: what is \(\frac{\partial}{\partial x} x^2\)?

If we mean to take \(\frac{\partial f}{\partial x} (x,x)\), then this is \[ \left.\frac{\partial f}{\partial x} (x,y) \right|_{(x,x)} = 2x\] On the other hand, if we mean \(\frac{\partial g}{\partial x} (x,x)\), then this is \[\left.\frac{\partial g}{\partial x} (x,y) \right|_{(x,x)} = 0\]

Folland gives other examples of when the partial derivative notation doesn’t make any sense (it can mean two different things).