I am unsure as to what notation is allowed and what is not, even though I can describe the the partial derivatives in words, I sometimes apparently get the notation incorrect. To fully compare and contrast, here are two questions, with two different function compositions, expressed in two different ways.
Question 1
Suppose $f:\mathbb{R}^2 \rightarrow \mathbb{R}^m$ and $g:\mathbb{R} \rightarrow \mathbb{R}^2$ .
Define $\vec{z}=\vec{\varphi}(u) = \left(\vec{f} \circ \vec{g}\right)(u)$ and let $s=\vec{f}\left(\left\langle x, y\right\rangle\right)$ and $x=\vec{g_1}(u)$ and $y=\vec{g_2}(u)$.
In this case, because it explicitly said that $f$ is a function of $x$ and $y$, we can say (correct me if I'm wrong):
$\frac{\partial f}{\partial u} =\frac{\partial f}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial u}$ (Expression 1)
Are the following also acceptable ways to write the same thing?
Replacing $f$ with $z$
$\frac{\partial z}{\partial u} =\frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u}$ (Expression 2)
or, replacing $f$ with $\varphi$
$\frac{\partial \varphi}{\partial u} =\frac{\partial \varphi}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial \varphi}{\partial y} \frac{\partial y}{\partial u}$ (Expression 3)
or, replacing $f$ with $f \circ g$
$\frac{\partial (f \circ g)}{\partial u} =\frac{\partial (f \circ g)}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial (f \circ g)}{\partial y} \frac{\partial y}{\partial u}$ (Expression 4)
or, replacing $x$ with $g_1$ and $y$ with $g_2$
$\frac{\partial f}{\partial u} =\frac{\partial f}{\partial g_1} \frac{\partial g_1}{\partial u} + \frac{\partial f}{\partial g_2} \frac{\partial g_2}{\partial u}$ (Expression 5)
or, replacing $f$ with $\varphi$ and $x$ with $f_1$ and $y$ with $f_2$
$\frac{\partial \varphi}{\partial u} =\frac{\partial \varphi}{\partial f_1} \frac{\partial f_1}{\partial u} + \frac{\partial \varphi}{\partial f_2} \frac{\partial f_2}{\partial u}$ (Expression 6)
Question 1: Are all these notations equivalent?
I'm not sure about Expression 5, $f$ is written as a function of $x$ and $y$, but $x=g_1$ and $y=g_2$, so I guess this is okay?
Question 2
In Question 1, it is explicitly written that $f$ is a function of $x$ and $y$.
But what if this is not the case? For example:
Let $f:\mathbb{R^{n}} \rightarrow \mathbb{R}$. Let $v \in \mathbb{R}^n$. Let $t$ be a scalar and $x \in \mathbb{R^n}$.
Let $\varphi: \mathbb{R^{n+1}} \rightarrow \mathbb{R} $ be a function defined as such: $\varphi(t,x) = f(2x+tv)$.
What valid ways are there to write the partial derivative of $\varphi$ with respect to $t$?
- Could I let $y=2x+tv$ and say $\frac{\partial \varphi}{\partial t} = \frac{\partial \varphi}{\partial y_1}\frac{\partial y_1}{\partial t} + \ldots + \frac{\partial \varphi}{\partial y_n}\frac{\partial y_n}{\partial t} $?
I've been told this is incorrect, since $f$ is not a function of $y$, and it should be $f$ like in #2:
Could I say $\frac{\partial \varphi}{\partial t} = \frac{\partial \varphi}{\partial f_1}\frac{\partial f_1}{\partial t} + \ldots + \frac{\partial \varphi}{\partial f_n}\frac{\partial f_n}{\partial t}$?
Could I let $g(t,x) = 2x+tv$ and say $\frac{\partial \varphi}{\partial t} = \frac{\partial f}{\partial t} = \frac{\partial \varphi}{\partial g_1}\frac{\partial g_1}{\partial t} + \ldots + \frac{\partial \varphi}{\partial g_n}\frac{\partial g_n}{\partial t}$?
For #3 here, if this is incorrect, how does this fundamentally differ from doing Expression 5 above?
- Could I let $g(t,x) = 2x+tv$ and say $\frac{\partial \varphi}{\partial t} = \frac{\partial f}{\partial t} = \frac{\partial f}{\partial g_1}\frac{\partial g_1}{\partial t} + \ldots + \frac{\partial f}{\partial g_n}\frac{\partial g_n}{\partial t}$?
Some classic books have $z=f(x,y)$ and functions $x=x(u,v)$ and $y=y(u,v)$, then, in appropriate conditions, holds you expression (2) $$\frac{\partial z}{\partial u} =\frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u}$$ To not mix variables with functions you can write $x=\phi(u,v)$ and $y=\xi(u,v)$, then it will be $$\frac{\partial f}{\partial u} =\frac{\partial f}{\partial x} \frac{\partial \phi}{\partial u} + \frac{\partial f}{\partial y} \frac{\partial \xi}{\partial u}$$
Now if you take $\varphi(t,x) = f(2x+tv)$, then let's write it as $z=f(y)$ and $y=g(t,x) = 2x+tv$. Then we have $$\frac{\partial \varphi}{\partial t}=\frac{\partial f}{\partial t} = \frac{d f}{d y}\frac{\partial g}{\partial t} = \frac{d f}{d y} \cdot v$$
We can consider, that $f$ have $n$ coordinates, $z=f(y_1, \cdots, y_n)$ and $y_i=\psi_i(x_1, \cdots, x_m)$ creates $\psi$.Then
$$\frac{\partial f \circ \psi}{\partial x_k}= \frac{\partial f}{\partial y_1}\frac{\partial \psi_1}{\partial x_k}+\frac{\partial f}{\partial y_2}\frac{\partial \psi_2}{\partial x_k}+ \cdots+\frac{\partial f}{\partial y_n}\frac{\partial \psi_n}{\partial x_k}$$