Notation in Spivak's Calculus on Manifolds

[Z90E532]

I have a question regarding the usage of notation on problem 2-11.

Find $f'(x, y)$ where $f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)$ where $h = \int ^t _a g$ and $g : R \rightarrow R$

Since no differential is given, what exactly are we integrating with respect to?

This looks like a composition of $h$ with some sort of identity operator matrix multiplied by $(x,y)$, but I'm not exactly sure how it works. I've never this notation used anywhere else.

 

[slider142]

$g$ is a single variable function, so this is an ordinary single variable integral. The name of the single variable of integration is not relevant to the problem: you can use any name you like.
The three notations $\int_1^2 g$, $\int_1^2 g(t) \, dt$, and $\int_1^2 g(x)\, dx$ all refer to exactly the same computation and the same number, if the definite integral exists.
$\pi_1$ and $\pi_2$ are the first and second coordinate projection functions. In particular, $\pi_1(x, y) = x$ and $\pi_2(x, y) = y$.
The notation $[h\circ (\pi_1 + \pi_2)](x, y)$ is not meant to be multiplication. It is meant to be functional notation, in the same way that $f(x)$ means that value of the function $f$ when the input is $x$, not $f$ multiplied by $x$. In your case, this is the function $h\circ (\pi_1 + \pi_2)$ applied to the input $(x, y)$.