Notation in Spivak's Calculus on Manifolds

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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.
 

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##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)##.
 

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