Partial / Total Derivative, Compositions

[SchroedingersLion]

Hello there,

I have stumbled across further examples to derivatives of multivariable functions that confuse me. Similar to my other thread:
https://www.physicsforums.com/threads/partial-derivative-of-composition.985371/#post-6309196

Suppose we have two functions, $f: R^2 \rightarrow R, (t,x) \mapsto f(t, x)$ and $g: R \rightarrow R, t \mapsto g(t)$ .

We have $$\frac {df} {dt} = \frac {\partial f} {\partial x}\frac {\partial x} {\partial t} + \frac {\partial f} {\partial t} .$$
If we now write $x=g(t)$ and consider $f(t, g(t))$ what do people actually mean by this?

Option A) I still view it as the function $f$, simply evaluated at $x=g(t)$.
Then $\frac {\partial } {\partial t} f(t, g(t)) = \frac {\partial f(t, x)} {\partial t}$, and $\frac {d} {dt} f(t, g(t))= \frac {\partial f} {\partial x}\frac {\partial x} {\partial t} + \frac {\partial f} {\partial t}$ as above.

Option B) I take $f(t, g(t))$ to be a partial composition, i.e. I have a new function $h(t)=f(t,g(t))$. In that case, partial and total time derivatives are equal and should also be equal to the total time derivative of the interpretation of A).

So seeing that the total derivatives are equal for both cases, the interpretation decides the outcome of the partial derivative. I would have guessed that Option B is actually "correct". In a simpler case: If I have $f(x)$ and $x(t)$ (all simple 1D functions of the reals), I would write $$\frac {\partial } {\partial t} f(x)=0 \\ \frac {d} {dt} f(x) = \frac {\partial f} {\partial x} \frac {\partial x} {\partial t} \\ \text{and } \frac {\partial } {\partial t} f(x(t)) = \frac {\partial } {\partial t} (f \circ x)(t) = \frac {\partial x} {\partial t} \frac {\partial } {\partial x} f(x) = \frac {d} {dt} f(x) $$

So I would always assume $f(x(t))$ to imply a composition. Yet I have seen authors that treated it still as $f(x)$. Is there something wrong in my understanding, or is there really room for ambiguity here?

edit:
Wikipedia seems to contradict itself:
" In this case, we are actually interested in the behavior of the composite function $f(x, y(x))$ . The partial derivative of $f$ with respect to $x$ does not give the true rate of change of $f$ with respect to changing $x$ because changing $x$ necessarily changes $y$."
If they viewed it as a composite function, then the partial derivative of that composite function should give the whole variation...
https://en.wikipedia.org/wiki/Total_derivative#Example:_Differentiation_with_direct_dependencies

 

[andrewkirk]

The solution to the ambiguity is to recognise that the symbol string $f(t,g(t))$ is not a function but a real number. $f$ and $g$ are functions, but not $f(t,g(t))$. So to talk about the 'partial derivative of $f(t,g(t)$ with respect to $t$' is simply a category error, akin to talking about the temperature of $f(t,g(t))$. Unfortunately, it is a category error that writers often make, and what they mean by it is anybody's guess.

As you correctly point out, there are two functions in which we could interest ourselves, from that symbol string. The first is the function $a_t:\mathbb R\to \mathbb R$ such that $a_t(y) = f(y,g(t))$.
The second is the function $b:\mathbb R\to \mathbb R$ such that $b(y) = f(y,g(y))$.

Part of the problem stems from the fact that the Leibniz notation that is most commonly used for partial derivatives, and which is used in the OP, is inherently ambiguous, because it specifies the variable of differentiation by name rather than by position (what number argument it is). See this Insights post for more on this problem and how to use notation that removes the ambiguity.

Of course, we can't change how other people write. The best we can do when reading text that has this ambiguity is to try to work out the meaning from the context. We make a best guess as to what they mean, marking the page where we made the guess, then proceed until we encounter a problem that might suggest our guess was wrong, in which case we backtrack and try another interpretation.

 

[Delta2]

Indeed there is an ambiguity here but I think most people (including me) and most books would take option B).

You have to be more explicit when you write $$\frac{\partial f(t,g(t))}{\partial t}$$ if you want to mean option A). You just have to say it with words I guess or write something like this maybe $$\left.{\frac{\partial f(t,x)}{\partial t}}\right|_{x=g(t)}$$

 

[SchroedingersLion]

The solution to the ambiguity is to recognise that the symbol string $f(t,g(t))$ is not a function but a real number. $f$ and $g$ are functions, but not $f(t,g(t))$. So to talk about the 'partial derivative of $f(t,g(t)$ with respect to $t$' is simply a category error, akin to talking about the temperature of $f(t,g(t))$. Unfortunately, it is a category error that writers often make, and what they mean by it is anybody's guess.

As you correctly point out, there are two functions in which we could interest ourselves, from that symbol string. The first is the function $a_t:\mathbb R\to \mathbb R$ such that $a_t(y) = f(y,g(t))$.
The second is the function $b:\mathbb R\to \mathbb R$ such that $b(y) = f(y,g(y))$.

Part of the problem stems from the fact that the Leibniz notation that is most commonly used for partial derivatives, and which is used in the OP, is inherently ambiguous, because it specifies the variable of differentiation by name rather than by position (what number argument it is). See this Insights post for more on this problem and how to use notation that removes the ambiguity.

Of course, we can't change how other people write. The best we can do when reading text that has this ambiguity is to try to work out the meaning from the context. We make a best guess as to what they mean, marking the page where we made the guess, then proceed until we encounter a problem that might suggest our guess was wrong, in which case we backtrack and try another interpretation.

Thank you andrewkirk for your input and for the great reference! Now I understand why mathematicians often prefer to use the $D_k$ notation. So, there really is ambiguity.
$\frac{\partial}{\partial t} f(t, x(t))$ then means either $D_1 f(t, x(t)) = \frac{\partial}{\partial t} f(t, x) |_{(t, x(t))}$ or $\frac{\partial}{\partial t} h(t)$ with $h(t)=f(t, x(t))$, which is equivalent to interpreting $f(t, x(t))$ as a partial composition of $f(t, x)$ and $x(t)$.

Indeed there is an ambiguity here but I think most people (including me) and most books would take option B).

You have to be more explicit when you write $$\frac{\partial f(t,g(t))}{\partial t}$$ if you want to mean option A). You just have to say it with words I guess or write something like this maybe $$\left.{\frac{\partial f(t,x)}{\partial t}}\right|_{x=g(t)}$$

Thank you Delta2. I would agree that B) sounds more reasonable, but then again in classical mechanics the Lagrangian is often written as $L(t, \mathbf{q}, \mathbf{p})$ where $\mathbf{q}=\mathbf{q(t)}$ and $\mathbf{p}=\mathbf{p(t)}$. The partial time derivative is supposed to act on the explicit $t$ only, so
$$\frac{\partial}{\partial t} L(t, \mathbf{q}, \mathbf{p}) = D_1 L(t, \mathbf{q}, \mathbf{p}). $$
If we auto-interpret $L(t, \mathbf{q}, \mathbf{p})$ as a single function of $t$, then we would get the wrong result. Unless you want to distinguish -which would make sense- between writing $L(t, \mathbf{q}, \mathbf{p})$ and $L(t, \mathbf{q}(t), \mathbf{p}(t)).$ But I don't think most authors make this distinction. So one really needs to consider the context...

 

[Delta2]

Unless you want to distinguish -which would make sense- between writing L(t,q,p)L(t,q,p) L(t, \mathbf{q}, \mathbf{p}) and L(t,q(t),p(t))

That's what makes the difference , at least for me. This means that we can treat t,q,p as being independent variables though they aren't actually independent.