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Now I realize there are a few questions like this on here, but none of them really get to the heart of what I'm asking...

In single variable calculus we learn that the following can describe the relationship between a derivative and an integral...

$$ \int \frac{dy}{dx} dx = \int dy \frac{dx}{dx} = \int dy = y $$

In multivariable calculus we learn to take partial derivatives using the $\partial$ symbol but then unintuitively learn to take double integrals with this notation... $$ \int\int f(x,y)dxdy $$ But using the relationships from single variable calculus (of a derivative to an integral), this would intuitively seem to be the way to notate a double integral...

$$\int\int \frac{\partial z}{\partial x \partial y} \partial x \partial y = \int\int \partial z \frac{\partial x}{\partial x } \frac{\partial y}{\partial y } = \int\int \partial z = z $$

As opposed to the common notation of... $$ f(x,y)=\frac{\partial z}{\partial x \partial y} \rightarrow \int\int f(x, y) dx dy = z $$ Wouldn't this, however, imply... $$\int\int \frac{\partial z}{\partial x \partial y} dx dy = \int\int \partial z \frac{dx}{\partial x } \frac{dy}{\partial y } $$

For what reason do we use this notation when it doesn't seem algebraically consistent like single variable notation?

Does $ \frac{dx}{\partial x} = 1 $ in the same way that $ \frac{dx}{dx} = 1 $ ?

Am I wildly overthinking things?

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  • $\begingroup$ I'm not sure if this is what you're intending for, but when we do the dx by itself it's treated as a single-variable function in terms of x (so $f(x, y)=g(x)$ and y is constant). Then after it's integrated like a single-variable integral it's treated as a function in terms of y, while still following single-variable rules. $\endgroup$ – pie314271 Nov 15 '16 at 1:03
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    $\begingroup$ The "algebraic consistency" of the single-variable $dx$ notation is not something that is totally obvious. The limits, pros and cons of algebraic manipulations of differentials are extensively discussed here: math.stackexchange.com/questions/21199/… $\endgroup$ – user159517 Nov 15 '16 at 1:05
  • $\begingroup$ @pie314271 but isn't that exactly what the $ \partial $ is supposed to specify? That's certainly seems to be its meaning in derivation. $\endgroup$ – DeathByTensors Nov 15 '16 at 1:06
  • $\begingroup$ @DrewBuckley and to answer your question, in my opinion you are right to say that one should actually write $\int\int f(x,y) \partial x \partial y$ as it would be consistent with the way the $df$, $\partial f$ notations are used for multivariate functions. But the convention is to write this way, so I'd advise to stick to the convention and don't think too much about it. $\endgroup$ – user159517 Nov 15 '16 at 1:09
  • $\begingroup$ @DrewBuckley The partial sign is only for multivariable functions; here functions like g(x) are treated as single-variable and therefore cannot have partial signs. True, conventions are a little strange sometimes. $\endgroup$ – pie314271 Nov 15 '16 at 1:20

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