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 four possibilities:

1. It's $f'$ (i.e., the total derivative of f) evaluated at the point (x,x). Once we fix a point $(x_{0},y_{0})$, then $f'(x_{0},y_{0})$ is a linear map $\mathbf {R} ^{2}\to \mathbf {R} ^{2}$ defined by the matrix ${\begin{pmatrix}2x_{0}&0\\0&2y_{0}\end{pmatrix}}$. Applying this linear map to (x,x), we get ${\begin{pmatrix}2x_{0}&0\\0&2y_{0}\end{pmatrix}}{\begin{pmatrix}x\\x\end{pmatrix}}={\begin{pmatrix}2x_{0}x\\2y_{0}x\end{pmatrix}}\in \mathbf {R} ^{2}$.
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). We have $(2x,0)+(0,2y){\frac {dy}{dx}}=(2x,2y{\frac {dy}{dx}})$. We can't evaluate this further since we don't know how x and y are related. If we assume the relationship y=x then this reduces to (2x,2x).
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 result is the function $x\mapsto (2x,2x):\mathbf {R} \to \mathbf {R} ^{2}$.
4. There's implicitly a function $\phi (x,y)=(x,x)$, so ${\frac {d}{dx}}f(x,x)={\frac {d}{dx}}f(\phi (x,y))$. Using the chain rule, this is $(f\circ \phi )'(x,y)=f'(\phi (x,y))\phi '(x,y)=\left.{\begin{pmatrix}2x_{0}&0\\0&2y_{0}\end{pmatrix}}\right\vert _{(x_{0},y_{0})=\phi (x,y)}{\begin{pmatrix}1&0\\1&0\end{pmatrix}}={\begin{pmatrix}2x&0\\2x&0\end{pmatrix}}$.

Big picture

Why is this notation so confusing? I think there are two (?) big reasons:

• The notation violates the substitution axiom of equality. We write things like z = z(x,y) where the same symbol z now has two different types. Folland's example of ${\frac {\partial w}{\partial t}}$ meaning two different things.
• The precedence of the differentiation operator is sometimes unclear. e.g. this is the case with $\nabla f(Ax)$ and ${\frac {d}{dx}}f(x,x)$.

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