Notation for partial derivatives

Question

I thought that the meaning of $$ \frac{\partial f(x, y, z)}{\partial x} $$ is differentiation on $x$ with fixed $y$ and $z$. So $(x, y, z)$ in the numerator is just saying which variables are fixed. If I need to indicate where the derivative is evaluated, I write it in the right of a vertical bar as a subscript. But today my teacher used $(x, y, z)$ in the numerator to denote where the derivative is evaluated. So, for example, $$ \frac{\partial f(0, 0, 0)}{\partial x} $$ means $$ \frac{\partial f(x, y, z)}{\partial x} \bigg\rvert_{x=0,y=0,z=0} $$ Is that a standard convention? If so, what is the meaning of this? $$ \frac{\partial f(x, y, g(x, y))}{\partial x} $$ I have two candidates. One is a partial derivative of the composition of $f$ and $g$ where $g$ has some fixed value, and the other is the partial derivative of $f$ on $x$ evaluated at $(x, y, g(x, y))$. I think the two are not the same.

Answer 1

Congratulations, you have met one of the worst ambiguities in mathematical notation!

Assume you have a function of two variables, $f \colon A \times B \to \mathbb{R}$, where $A$ and $B$ are subsets of $\mathbb{R}$. The notation $$\frac{\partial f}{\partial x}(x_0,y_0)$$ is commonly used to denote the value of the partial derivative of $f$ with respect to the first variable, evaluated at $(x_0,y_0)$. This is the cleanest use of the notation for partial derivatives.

Anyway, it sometimes happens to use some lazy piece of notation such as $$\frac{\partial f(x,g(x,y))}{\partial x}$$ to denote the partial derivative of the map $(x,y) \mapsto f(x,g(x,y))$. This is imcompatible (in general) with the interpretation of the same formula as

The derivative of $f$ with respect to the first variable, evaluated at the point $(x,g(x,y))$.

This is bad, but it seems we have to live with it. Why? Just spend a couple of minutes and think about the second interpretation. To be rigorous, we should have written $$ \frac{\partial}{\partial x} \left( f \circ \left( (x,y) \mapsto (x,g(x,y)) \right) \right) (x,y), $$ which is a true nightmare.

Answer 2

The first interpretation is not correct, since there is no way to give $g$ some fixed value if it depends on $x$. The second interpretation is correct but worded in a confusing way, so let me clarify.

Consider the function $h(x,y)=f(x,y,g(x,y))$. Then $$ \frac{\partial f(x,y,g(x,y))}{\partial x}=\frac{\partial h(x,y)}{\partial x}. $$ The result is a function of two variables, call it $k(x,y)$. To calculate e.g. $k(x,5)$ you would compute $$ k(x,5)=\frac{d}{dx}f(x,5,g(x,5)). $$ This is a derivative in the usual sense, not a partial derivative any more. By the chain rule, we can calculate $k(x,5)$ as follows: $$ k(x,5)=f_x(x,5,g(x,5)) + f_z(x,5,g_x(x,5)). $$ There is nothing special about the number $5$, it could be anything. So replacing $5$ with $y$, we get the formula $$ \frac{\partial f(x,y,g(x,y))}{\partial x}=f_x(x,y,g(x,y)) + f_z(x,y,g_x(x,y)). $$

Answer 3

1. Yes, $\frac{\partial f(x,y,z)}{\partial x}$ is derivative w.r.t. $x$ at fixed $y,z$.
2. $\frac{\partial f(0,0,0)}{\partial x}$ is not standard notation. Strictly speaking, it should be zero, because $f(0,0,0)$ is a constant which does not depend on $x$. Sometimes, yes, it is used as a shorthand for $\frac{\partial f(x,y,z)}{\partial x}|_{x=y=z=0}$. But you should only do that if it is extreamly clear from the context what you mean. Generally, avoid this notation.
3. The third expression is a derivative w.r.t. $x$, but $x$ appears two times in the numerator. Using standard derivation-rules you can compute $$\frac{\partial f(x,y,g(x,y))}{\partial x} = \frac{\partial f(x,y,z)}{\partial x}\bigg\rvert_{z=g(x,y)} + \frac{\partial f(x,y,z)}{\partial z}\bigg\rvert_{z=g(x,y)} \cdot \frac{\partial g(x,y)}{\partial x}$$

Answer 4

I think Siminore's answer is good. But I checked some textbooks just for curiosity.

1. "Advanced calculus" by Folland uses the notation like $\partial_x f(0, 0)$. Of course the meaning is the partial derivative of $f$ w.r.t. $x$ evaluated at $(0, 0)$. It does not use the notation $\partial f(0, 0) / \partial x$ extensively but there is a comment that you can use the notation.
2. "Advanced calculus" by Kaplan and "The way of analysis" by Strichartz also follow the same convention.
3. "Real mathematical analysis" by Pugh uses the notation $\partial f(0, 0) / \partial x$ and in some place it uses $\partial f(x, g(x)) / \partial x$ to denote the partial derivative of $f$ w.r.t. $x$ evaluated at $(x, g(x))$. It doesn't mean the derivative of the composite function.
4. I also checked "Principles of mathematical analysis" by Rudin. It looks like to avoid the notation $\partial f(a, b) / \partial x$. Instead it says "$\partial f / \partial x$ at $(a, b)$". However, actually I found only one such occurrence. There is not much use of the round notation.
5. "Mathematical methods for physicists" by Afken uses the notation $\partial f(x, 0)/\partial t$ to denote $\partial f(x, t)/\partial t |_{t = 0}$. It uses the two notations interchangeably.

Thus, the use case of your teacher is fairly common.

Answer 5

Seems like no one has mentioned that there is actually a totally unambiguous notation to deal with this problem, even though it is not in very common use:

$(\partial_1 f)(a,b,c) = \left. \dfrac{\partial f(x,y,z)}{\partial x} \right|_{(x,y,z)=(a,b,c)}$.

The "$1$" here indicates that $f$ is differentiated with respect to the first parameter. This is much better than using something like "$\partial_x$" where the $x$ is often used as a variable too and hence cannot serve well to indicate which parameter $f$ is differentiated with respect to.