Is it proper to write $\int \partial x$

Question

For single variable function, you write $\int dx$

But for multivariable function, can you write $\int \partial x$??

I've never seen the latter, can someone explain why?

Answer 1

$df$ is the total and $\partial f$ the partial differtial.

for a $$f(x,y)$$

you can write $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$ Sometimes you read $$ df= \partial_x dx+\partial_y dy$$

Never seen that someone "integrates" the $\partial x$. Alway $dx$

Answer 2

well, there is a reason to write $\int dx$, in fact when you are integrating you are using a measure, and if your function is a positive function, in fact you are measuring the area under the graph, and the properly notation is always dx, if you have multivalued functions depending on $x_1,\cdots ,x_n$ the you writre $\int f(x_1,\cdots ,x_n)dx_1\cdots dx_n$. Integral is a measure this is the reason.

Answer 3

It can be useful to write $ \int \partial x $ for indefinite integration. For example, if you're trying to antidifferentiate a differential form such as $ ( 2 x + y ) \, d x + ( x + 4 y ) \, d y $, then you can try doing partial antidifferentiation: $$ \int ( 2 x + y ) \, \partial x = x ^ 2 + y x + C _ x ( y ) $$ and $$ \int ( x + 4 y ) \, \partial y = x y + 2 y ^ 2 + C _ y ( x ) \text , $$ then inspect these to find the total antiderivative $$ \int ( 2 x \, d x + y \, d x + x \, d y + 4 y \, d y ) = x ^ 2 + x y + 2 y ^ 2 + C \text , $$ which matches both of the partial antiderivatives.

As in partial differentiation, so here in partial antidifferentiation, the use of $ \partial $ instead of $ d $ indicates that the other variables are to be treated as constants (part of which means that the constant of integration can depend on them).

But most people would just write $ d $ here instead of $ \partial $. This is a calculation in a limited context, and you'd probably write some words around it (or if you're just doing scratch work, at least think in your mind) saying that the other variable is constant, so you don't need the notation to remind you. It's different with differentiation, where a partial derivative might appear in a long calculation involving long expressions, where there is much more danger of forgetting what is being held constant and when.

You could make a case for doing the same even for definite integration in the case of an iterated integral. So instead of $$ \int _ a ^ b \int _ c ^ d f ( x , y ) \, d y \, d x \text , $$ you could write $$ \int _ a ^ b \int _ c ^ d f ( x , y ) \, \partial y \, d x \text , $$ to emphasize that when you do the inner integral with respect to $ y $, you're keeping $ x $ constant. But that's also kind of obvious, because of the integral with respect to $ x $ immediately around it! So while I have seen $ \partial $ used (very rarely) with indefinite integrals, I've never actually seen this done with a definite integral.