why does multivariable integration not work?

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d(PV)=VdP+PdV
Int(dPV)=int(VdP+PdV) != Int(VdP)+ int(PdV)

Why can you not distribute the integral?

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What does int(VdP+PdV) mean if not int(VdP) + int(PdV)? If you wanted to “integrate them together” you’d have to divide and multiply by some other differential:

VdP = V(dP/dt)dt, PdV= P(dV/dt)dt

int((VdP/dt + PdV/dt)dt)

You can’t integrate with respect to two different things at once, what would the limits be?

Imagine you’ve got a 5×5 grid of penny stacks. Each stack is a different height and they’re laid out in those 25 squares.

Now, the question becomes: how many total pennies do you have?

You can’t simply pick one row or one column to count them all, can you? You’d be missing the count for all the other rows and columns.

What you have to do is, for each row (or column), count up all the pennies in that row (or column) before moving to the next one. Then you count in the next one. And so on until you get to the end.

This is ‘multi-variable integration’ and it’s done by nesting integrations rather than merely adding them.