What is wrong with this chain rule calculation?

Question

I came across this case of a derivative: $v=x-t,w=t$ and a function $u$. So using the chain rule I get: $$ \frac{\partial u}{\partial x}= \frac{\partial u}{\partial v}\frac{\partial v}{\partial x}+\frac{\partial u}{\partial w}\frac{\partial w}{\partial x}=\frac{\partial u}{\partial v} $$ $$ \frac{\partial u}{\partial t}= \frac{\partial u}{\partial v}\frac{\partial v}{\partial t}+\frac{\partial u}{\partial w}\frac{\partial w}{\partial t}=-\frac{\partial u}{\partial v}+\frac{\partial u}{\partial w}. $$ This seems incorrect to me since $w=t$ so $\dfrac{\partial u}{\partial w}=\dfrac{\partial u}{\partial t}$ which is not what I get. But what is wrong?

Answer 1

What does $\partial u/\partial w$ mean? If you write $u$ as a function of $v=x-t$ and $w=t$ and keep $v$ constant, we take the limit of $\Delta u/\Delta w$ as $\Delta w\to 0$ along a curve (well, line) of constant $v$. On the other hand, $\partial u/\partial t$ is defined as the limit of $\Delta u/\Delta t$ along a curve of constant $x$. In other words, in the $xt$-plane with coordinates $(x,t)$, the partial $\partial u/\partial t$ represents the directional derivative in the positive $t$-direction (along horizontal lines of constant $x$), whereas $\partial u/\partial w$ represents the directional derivative along lines of slope $1$ (i.e. along diagonal lines of constant $v$).

So the partials are definitionally different: the partial derivative's definition depends not just on the variable you see explicitly written but also implicitly on the other variable that goes with it in the coordinate system and which is kept constant. Specifying a single coordinate (recall a coordinate is formally a scalar function defined on an abstract space, such as $(x,t)\mapsto t$) is technically not enough to even define a partial derivative: you need two coordinates for a 2D domain.

In thermodynamics (e.g.), where the state space of a system admits many different coordinate systems useful for different contexts, there is notation to specify what variables (or coordinates) are kept constant: subscripts. In this setting, we don't need the extra letter $w$ for $t$ but we can set $v=x-t$. Then the partials are denoted by $(\partial u/\partial t)_x$ and $(\partial u/\partial t)_v$, which highlights that even though both are with respect to $t$, the variable which is kept constant differs between the two so there is no expectation they should be equal.

Answer 2

The two partial derivatives are not always equal. While $w=t$, their is no equality for the covariables $v$ and $x$.

The first is the partial derivative of $u$ with respect to $w$ while implicitly keeping $v$ constant, while the second is the partial derivative of $u$ with respect to $t$ while implicitly keeping $x$ constant.

Consider the following example:

$\begin{align}\textrm{Let }u &= v^2+w^2\\ \partial u/\partial w&=2w\\u&=(x-t)^2+t^2&&\textrm{by substitution}\\ \partial u/\partial t&= -2(x-t)+2t\\&=4t-x\end{align}$

See also the more in-depth answer by @runway44 .