Mathematics and notation
Last substantive revision: 2015-10-29
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).
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