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 $\mathrm{\nabla }f(Ax)$
• 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 $f^{\prime }$ (i.e., the total derivative of f) evaluated at the point (x,x). Once we fix a point $(x_0,y_0)$, then $f^{\prime }(x_0,y_0)$ is a linear map ${\mathbf{R}}^2\to {\mathbf{R}}^2$ defined by the matrix $\left(\begin{array}{cc}2x_0& 0\\ 0& 2y_0\end{array}\right)$. Applying this linear map to (x,x), we get $\left(\begin{array}{cc}2x_0& 0\\ 0& 2y_0\end{array}\right)\left(\begin{array}{c}x\\ x\end{array}\right)=\left(\begin{array}{c}2x_{0}x\\ 2y_{0}x\end{array}\right)\in {\mathbf{R}}^2$.
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 $\varphi (x,y)=(x,x)$, so $\frac{d}{dx}f(x,x)=\frac{d}{dx}f(\varphi (x,y))$. Using the chain rule, this is $(f\circ \varphi {)}^{\prime }(x,y)=f^{\prime }(\varphi (x,y)){\varphi }^{\prime }(x,y)={\left(\begin{array}{cc}2x_0& 0\\ 0& 2y_0\end{array}\right)|}_{(x_0,y_0)=\varphi (x,y)}\left(\begin{array}{cc}1& 0\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}2x& 0\\ 2x& 0\end{array}\right)$.

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 $\mathrm{\nabla }f(Ax)$ 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