The following problem is from Chapter 8 "Differential Calculus of Scalar and Vector Fields" from Apostol's Calculus, Volume II.
- The equations $u=f(x,y), x=X(s,t), y=Y(s,t)$ define $u$ as a function of $s$ and $t$, say $u=F(s,t)$.
a) Use an appropriate form of the chain rule to express the partial derivatives $\frac{\partial F}{\partial s}$ and $\frac{\partial F}{\partial t}$ in terms of $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial X}{\partial s}, \frac{\partial X}{\partial t}, \frac{\partial Y}{\partial s}, \frac{\partial Y}{\partial t}$.
b) If $\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2 f}{\partial y \partial x}$, show that
$$\frac{\partial^2 F}{\partial s^2}=\frac{\partial f}{\partial x}\frac{\partial^2 X}{\partial s^2}+\frac{\partial^2 f}{\partial x^2}\left ( \frac{\partial X}{\partial s} \right )^2+2\frac{\partial X}{\partial s}\frac{\partial Y}{\partial s}\frac{\partial^2 f}{\partial x \partial y}+\frac{\partial f}{\partial y}\frac{\partial^2 Y}{\partial s^2}+\frac{\partial^2 f}{\partial y^2}\left ( \frac{\partial Y}{\partial s} \right )^2$$
My question is about item b). Let me go through item a).
We have
$$F(s,t)=F(X(s,t), Y(s,t))$$
We don't know if $f$ and $F$ are scalar or vector fields.
Assume they are scalar fields. Then the total derivative of $F$ is
$$DF(s,t)=D[f(g(s,t))]=Df(g(s,t))Dg(s,t)$$
where $D$ means "Jacobian matrix of".
Hence we have
$$\begin{bmatrix} \frac{\partial F}{\partial s} & \frac{\partial F}{\partial t} \end{bmatrix}=\begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \end{bmatrix}\begin{bmatrix} \frac{\partial X}{\partial s} & \frac{\partial X}{\partial t} \\ \frac{\partial Y}{\partial s} & \frac{\partial Y}{\partial t}\end{bmatrix}$$
which is what we are looking for in item a).
Now let's try to compute $\frac{\partial^2 F}{\partial s^2}$.
We start with what we found in a), namely
$$\frac{\partial F}{\partial s}=\frac{\partial f}{\partial x}\frac{\partial X}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial Y}{\partial s}$$
and we take the total derivative again
$$\begin{bmatrix} \frac{\partial^2 F}{\partial s^2} & \frac{\partial^2 F}{\partial s\partial t} \end{bmatrix}=\begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\partial y} \end{bmatrix}\begin{bmatrix} \frac{\partial X}{\partial s} & \frac{\partial X}{\partial t} \\ \frac{\partial Y}{\partial s} & \frac{\partial Y}{\partial t}\end{bmatrix}\frac{\partial X}{\partial s}+\frac{\partial f}{\partial x}\begin{bmatrix} \frac{\partial^2 X}{\partial s^2} & \frac{\partial^2 X}{\partial s\partial t} \end{bmatrix}$$
$$+\begin{bmatrix} \frac{\partial^2 f}{\partial y^2} & \frac{\partial^2 f}{\partial y\partial x} \end{bmatrix}\begin{bmatrix} \frac{\partial X}{\partial s} & \frac{\partial X}{\partial t} \\ \frac{\partial Y}{\partial s} & \frac{\partial Y}{\partial t}\end{bmatrix}\frac{\partial Y}{\partial s}+\frac{\partial f}{\partial y}\begin{bmatrix} \frac{\partial^2 Y}{\partial s^2} & \frac{\partial^2 Y}{\partial s\partial t} \end{bmatrix}$$
I used some sort of "product rule" above that I am extrapolating from the single-variable product rule. I am not sure this rule is true here, and I did not see it in the book yet. So maybe it's just wrong to use it.
This comes out to
$$\frac{\partial^2 F}{\partial s^2}=\frac{\partial^2 f}{\partial x^2}\left ( \frac{\partial X}{\partial s} \right )^2 + \frac{\partial^2 f}{\partial x\partial y}\frac{\partial Y}{\partial s}\frac{\partial X}{\partial s}+\frac{\partial f}{\partial x}\frac{\partial^2 X}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\frac{\partial X}{\partial s}\frac{\partial Y}{\partial s}+\frac{\partial^2 f}{\partial y\partial x}\left (\frac{\partial Y}{\partial s}\right )^2+\frac{\partial f}{\partial y}\frac{\partial^2 Y}{\partial s^2}$$
which isn't quite the expected result (at least it seems not to be).
My main goal is to make sure the intermediate steps are conceptually correct. Of course if they are then the answer should come out correct. So my question is if the derivatives above are all being taken in the correct manner?
Turns out there is a mistake in equation (1) in the original post.
It should be
$$\begin{bmatrix} \frac{\partial^2 F}{\partial s^2} & \frac{\partial^2 F}{\partial s\partial t} \end{bmatrix}=\begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x\partial y} \end{bmatrix}\begin{bmatrix} \frac{\partial X}{\partial s} & \frac{\partial X}{\partial t} \\ \frac{\partial Y}{\partial s} & \frac{\partial Y}{\partial t}\end{bmatrix}\frac{\partial X}{\partial s}+\frac{\partial f}{\partial x}\begin{bmatrix} \frac{\partial^2 X}{\partial s^2} & \frac{\partial^2 X}{\partial s\partial t} \end{bmatrix}$$
$$+\begin{bmatrix} \frac{\partial^2 f}{\partial y\partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}\begin{bmatrix} \frac{\partial X}{\partial s} & \frac{\partial X}{\partial t} \\ \frac{\partial Y}{\partial s} & \frac{\partial Y}{\partial t}\end{bmatrix}\frac{\partial Y}{\partial s}+\frac{\partial f}{\partial y}\begin{bmatrix} \frac{\partial^2 Y}{\partial s^2} & \frac{\partial^2 Y}{\partial s\partial t} \end{bmatrix}\tag{1}$$
Note the first matrix on the second row above
$$\begin{bmatrix} \frac{\partial^2 f}{\partial y\partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}$$
In the original post it was
$$\begin{bmatrix} \frac{\partial^2 f}{\partial y^2} & \frac{\partial^2 f}{\partial y\partial x} \end{bmatrix}$$
Making this correction, the correct result is obtained.