My physics textbook presented me with the following relationship:
$$\left( \dfrac{\partial x}{\partial t} \right)_\varphi = \dfrac{-(\partial \varphi / \partial t)_x}{(\partial \varphi / \partial x)_t}$$
In researching this relationship, I encountered this Wikipedia article on the triple product rule. The article says the following:
The triple product rule for such interrelated variables $x$, $y$, and $z$ comes from using a reciprocity relation on the result of the implicit function theorem and is given by
$$\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1.$$
Note: In each factor the variable in the numerator is considered to be an implicit function of the other two. In each factor the subscripted variable is being held constant.
Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of $x$ with respect to $y$ with $z$ held constant, one would write $x$ as a function of $y$ and $z$ and take the partial derivative of this function with respect to $y$ only.
The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,
$$\left({\frac {\partial x}{\partial y}}\right)_{z}=-{\frac {\left({\frac {\partial z}{\partial y}}\right)_{x}}{\left({\frac {\partial z}{\partial x}}\right)_{y}}}$$
However, I'm unsure of how $\left({\frac {\partial x}{\partial y}}\right)_{z}=-{\frac {\left({\frac {\partial z}{\partial y}}\right)_{x}}{\left({\frac {\partial z}{\partial x}}\right)_{y}}}$ was derived from $\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}\left({\frac {\partial z}{\partial x}}\right)_{y}=-1$, unless there was some, what seems to me to be, abuse of notation? If we proceed as is typical, we get
$$\begin{align} \left({\frac {\partial x}{\partial y}}\right)_{z} \left({\frac {\partial y}{\partial z}}\right)_{x} = \dfrac{-1}{\left({\frac {\partial z}{\partial x}}\right)_{y}} \\ \Rightarrow \left({\frac {\partial x}{\partial y}}\right)_{z} &= \dfrac{ \left( \dfrac{-1}{\left({\frac {\partial z}{\partial x}}\right)_{y}} \right) }{\left({\frac{\partial y}{\partial z}}\right)_{x}} \\ &= \dfrac{-1}{\left({\frac{\partial z}{\partial x}}\right)_{y} \left({\frac{\partial y}{\partial z}}\right)_{x}} \end{align}$$
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
I think it's the other way around. That is, the triple product formula follows from a more basic analysis. Namely let $(x,y,z)\in \mathbb R^3$ and suppose $F:\mathbb R^3\to \mathbb R$ with $F(x,y,z)=0$. Then, if none of the partials vanish at $(x,y,z)$, the implicit function theorem provides functions $f,g,h$ such that $x=f(y,z),\ y=g(x,z),\ z=h(x,y),$ which are differentiable, at least in small neighborhoods contained in their domains.
Then, the matrix representation of the (total) derivative of the composition $F(f(y,z),y,z)$ is given by
$$\begin{pmatrix} F_x &F_y &F_z \end{pmatrix}\begin{pmatrix} f_y & f_z\\ 1 & 0 \\ 0&1 \end{pmatrix}=\begin{pmatrix} 0& 0 \end{pmatrix}$$
and then $F_x\cdot f_y+F_y\cdot 1=0$ and $F_x\cdot f_z+F_z\cdot 1=0$ from which we obtain the familiar formulas
$$\frac{\partial x}{\partial y}=f_y=-\frac{F_y}{F_x}\quad \text{and}\quad \frac{\partial x}{\partial z}=f_z=-\frac{F_z}{F_x}.$$
The other well-known formulas are obtained in the same way using $g$ and $h$ and then the triple product follows immediately by multiplying the appropriate formulas. For example
$$\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=\left(-\frac{F_y}{F_x}\right)\left(-\frac{F_z}{F_y}\right)\left(-\frac{F_x}{F_z}\right)=-1.$$