Let $f : \mathbb{R}^2 \to \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ both be scaler-valued differentiable functions such that $f(x,y) = u$ and $g(u) = z$. Therefore, we can construct the function $h = g \circ f$ such that $h : \mathbb{R}^2 \to \mathbb{R}$. If I wanted to apply the $\nabla$ operator to the function $h$, writing out the differentiation in Leibniz notation, would the following be correct? $$\nabla h = \frac{dg}{df}\frac{\partial f}{\partial x}dx + \frac{dg}{df}\frac{\partial f}{\partial y}dy$$
What I am concerned is incorrect is the use of the $d$'s and $\partial$'s. Would $$\nabla h = \frac{\partial g}{\partial f}\frac{\partial f}{\partial x}dx + \frac{\partial g}{\partial f}\frac{\partial f}{\partial y}dy$$
be the correct notation? If I am overcomplicating this procedure, please do not hesitate to share a simpler method.
Best Answer
$\nabla h$ is a covector with two components, so the $+$ in your first equation should be a comma:
$$\nabla h = \left(\frac{dg}{df}\frac{\partial f}{\partial x} , \frac{dg}{df}\frac{\partial f}{\partial y}\right)$$
Alternatively, you can write it with a + in the following way:
$$\nabla h = \frac{dg}{df}\frac{\partial f}{\partial x}dx+ \frac{dg}{df}\frac{\partial f}{\partial y}dy$$
$dx,dy$ denote the covectors $(1,0), (0,1)$ respectively.