Integral Notation: Abuse or Necessity?

[Mandelbroth]

Quick question:

When we write an integral $\displaystyle \int\limits_U f(x^1,\dots, x^n)~dx^1\wedge\cdots\wedge dx^n$, we really mean $\displaystyle \int\limits_U f\wedge dx^1\wedge\cdots\wedge dx^n$, right?

 

[rubi]

Both notations are right. The first one is already simplified, whereas the second one can be simplified further by explicitely computing the exterior product and you will end up with the first version. So I actually think the first version is cleaner.

The differential form in your integral is an assignment of an element of $\Lambda(T^*_x U)$ to every point $x$ of $U$. $\Lambda(T^*_x U)$ is a vector space and $f(x)$ is just a coefficient (which is different at every point).

 

[Mandelbroth]

Both notations are right. The first one is already simplified, whereas the second one can be simplified further by explicitely computing the exterior product and you will end up with the first version. So I actually think the first version is cleaner.

The differential form in your integral is an assignment of an element of $\Lambda(T^*_x U)$ to every point $x$ of $U$. $\Lambda(T^*_x U)$ is a vector space and $f(x)$ is just a coefficient (which is different at every point).

I'm sorry, I should have specified what I mean.

I'm talking mainly about the fact that $f(x)$ is the output of $f$, whereas $f$ is the function.

 

[rubi]

I'm sorry, I should have specified what I mean.

I'm talking mainly about the fact that $f(x)$ is the output of $f$, whereas $f$ is the function.

In that case I would write $f \,\mathrm d x_1\wedge\ldots\wedge \mathrm d x_n$, however, omitting the first wedge. If you know that $f$ is a 0-form, then you can simplify the exterior product. But you are right: Inserting the coordinates into f gives you the form at a point already, whereas the form itself should be written without inserting $x$ into $f$.

 

[blue_raver22]

I believe that integral notation is a necessity in mathematics. It allows us to easily represent and manipulate complex mathematical concepts, such as integrals, in a concise and efficient manner. While it may seem like abuse to some, it is actually a powerful tool that allows us to solve a wide range of problems in various fields including physics, engineering, and economics.

The notation $\displaystyle \int\limits_U f(x^1,\dots, x^n)~dx^1\wedge\cdots\wedge dx^n$ is a shorthand representation of the more precise notation $\displaystyle \int\limits_U f\wedge dx^1\wedge\cdots\wedge dx^n$, where the function f is integrated over the region U. This notation makes it easier to perform calculations and understand the concept of integration without getting bogged down in notations and symbols.

In fact, the use of integral notation has been crucial in the development of many mathematical theories and has greatly advanced our understanding of the world around us. Without it, many important discoveries and advancements in fields such as calculus, physics, and economics would not have been possible.

Therefore, while it may seem like an abuse of notation to some, integral notation is a necessary and valuable tool in mathematics and science. It allows us to express complex ideas and solve difficult problems in a concise and efficient manner, making it an essential part of scientific research and discovery.