Notational confusion of multivariable derivatives

I think there's several different confusions that arise from multivariable derivative notation:

• The thing where ${\frac {\partial w}{\partial t}}$ can mean two different things on LHS and RHS when $t$ is used as both an initial and intermediate variable. (See Folland for details.)
• The thing where if $f(x,y)=(x^{2},y^{2})$ then ${\frac {\partial f}{\partial x}}(x,x)$ feels like it might be $(2x,2x)$ even though it's actually $(2x,0)$. (Example from Tao.) See also [1]
• The ambiguity of expressions like $\nabla f(Ax)$
• dual basis stuff -- see Tao's explanation of this in p. 225 of [2]

Working off the example from Tao above, let $f(x,y)=(x^{2},y^{2})$. What does ${\frac {d}{dx}}f(x,x)$ mean? Here are three possibilities:

1. It's $f'$ (i.e., the total derivative of f) evaluated at the point (x,x).
2. It's ${\frac {df}{dx}}$ (i.e., the total derivative of f with respect to x, in which we treat the variable y as a function of x, and use the chain rule ${\frac {df}{dx}}={\frac {\partial f}{\partial x}}+{\frac {\partial f}{\partial y}}{\frac {dy}{dx}}$ to compute) evaluated at the point (x,x).
3. We first compute f(x,x) to get an expression involving only x, which implicitly defines a function $\mathbf {R} \to \mathbf {R} ^{2}$. We now differentiate this function.

The derivative as a linear transformation in the several variable case and a number in the single-variable case

• The thing where the total derivative for $n=m=1$ "should" be a function but people treat it as a number. Refer to "Appendix A: Perorations of Dieudonne" (p. 337) in Pugh's Real Mathematical Analysis.

Total derivative versus derivative matrix

Technically the total derivative at a point is a linear transformation, whereas the derivative matrix is a matrix so an array of numbers arranged in a certain order. However, there is a one-to-one correspondence between linear transformations $\mathbf {R} ^{n}\to \mathbf {R} ^{m}$ and $m$ by $n$ matrices, so many books call the total derivative a matrix or equate the two.

A similar confusion exists in the teaching of linear algebra, where sometimes only matrices are mentioned.

See also

• Summary table of multivariable derivatives