Partial derivatives with a variable held constant

Question

Consider the cylindrical coordinate system $(\rho,\theta, z)$. If we define a new variable $\zeta = \theta - hz$, is there any way of rigorously defining the following partial derivative? $$ X=\frac{\partial}{\partial z}\bigg|_{\zeta,\rho} $$ Where $|_{\zeta,\rho}$ indicates that we are taking the derivative while holding the variables $\zeta$ and $\rho$ constant. This concept doesn't seem rigorously defined. Thinking of $X$ as a vector field, I imagine that it is the same as the projection of the constant unit vector field $\partial/\partial z$ onto the tangent spaces of the surfaces $\zeta=\text{const}$ and $\rho=\text{const}$, which would lead me to the conclusion that $$ \frac{\partial}{\partial z}\bigg|_{\zeta,\rho} = \frac{\partial}{\partial z}\bigg|_{\theta,\rho} + \frac{\partial}{\partial \theta}\bigg|_{z,\rho} $$ My main question is: How can I define the concept of 'holding a variable constant'?

Answer 1

I suggest that if you want to think clearly about this you oughtn't to introduce "a new variable", you ought to introduce a new triple of variables. The partial derivative of a function wrt any one of a triple of variables is well-defined, there's no need to keep repeating the rather confusing "keeping the other two fixed".

I also suggest that suppression of the function(s) involved is another source of confusion.

Suppose, then, that you are interested in a function $f(\rho,\theta,z)$. Suppose that each of these variables can be expressed in terms of a new triple of variables $R,\Theta,Z$: explicitly $\rho=\rho(R,\Theta,Z)$, $\theta=\theta(R,\Theta,Z)$ and $z=z(R,\Theta,Z)$.

Note: You are interested in the case $\rho(R,\Theta,Z)=R$, $\theta(R,\Theta,Z)=\Theta+hZ$, $z(R,\Theta,Z)=Z$.

Now the change of variables means that we have a new function

$$ F(R,\Theta,Z):=f(\rho(R,\Theta,Z),\theta(R,\Theta,Z),z(R,\Theta,Z)). $$

Note: You are interested in the partial derivative of $F$ with respect to the third variable, $Z$.

By the Chain rule we have: $$ \frac{\partial F}{\partial Z} = \frac{\partial f}{\partial \rho}\frac{\partial \rho }{\partial Z} + \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial Z} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial Z} . $$

Note: In your case $$ \frac{\partial \rho }{\partial Z}=0, \frac{\partial \theta}{\partial Z}=h, \frac{\partial z}{\partial Z}=1 $$ and so $$ \frac{\partial F}{\partial Z} = h\frac{\partial f}{\partial \theta} + \frac{\partial f}{\partial z} . $$

As I say, I think that it is confusing to suppress the functions -- the function of the left is not the function of the right -- and I also think that embroidering the partials with other symbols is confusing. (I know it is done in the old-fashioned books.)