How come surface integrals are single integrals in my book?

[Andrea94]

I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals.

For instance, they symbolically represent the magnetic flux through a surface as:
$$\int \vec{\textbf{B}} \cdot d\vec{\textbf{A}}$$
However, I suspected that this should in fact be a double integral (since the domain of integration is a surface), and indeed on Wikipedia they write the magnetic flux through a surface as:
$$\iint\vec{\textbf{B}} \cdot d\vec{\textbf{A}}$$

My question is, which representation is the right and why? Are they both right and we are supposed to implicitly understand that the single integral should be evaluated as a double integral since we have a surface area element?

 

[Mark44]

I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals.

For instance, they symbolically represent the magnetic flux through a surface as:
$$\int \vec{\textbf{B}} \cdot d\vec{\textbf{A}}$$
However, I suspected that this should in fact be a double integral (since the domain of integration is a surface), and indeed on Wikipedia they write the magnetic flux through a surface as:
$$\iint\vec{\textbf{B}} \cdot d\vec{\textbf{A}}$$

My question is, which representation is the right and why? Are they both right and we are supposed to implicitly understand that the single integral should be evaluated as a double integral since we have a surface area element?

I believe that both are correct, and that the first one you showed is an alternate notation for the second.

The wiki page on Multiple Integrals has this to say:

https://en.wikipedia.org/wiki/Multiple_integral][/PLAIN]
If f is Riemann integrable, S is called the Riemann integral of f over T and is denoted
$$\int \dots \int_T f(x_1, x_2, \dots, x_n) dx_1 \dots dx_n$$

Frequently this notation is abbreviated as
$$\int_T f(x)d^nx$$

where x represents the n-tuple (x1, ... xn) and dnx is the n-dimensional volume differential.

In your first integral, dA is akin to d²x in the integral above. In both cases they refer to a two-dimensional area differential.

 

[Andrea94]

I believe that both are correct, and that the first one you showed is an alternate notation for the second.

The wiki page on Multiple Integrals has this to say:

In your first integral, dA is akin to d²x in the integral above. In both cases they refer to a two-dimensional area differential.

That makes sense, thanks!

 

[Stephen Tashi]

I am currently reading Young & Freedmans textbook on physics as part of a university course, and I've noticed that they repeatedly represent surface integrals (which are double integrals) as single integrals.

From the viewpoint of pure mathematics, the definition of a surface integral does not define such an integral as a computation involving a double integral. I suspect the rigorous definition of a surface integral is so complicated that it is rarely seen in physics texts, but usually there is some attempt in physics texts to define it without making reference to iterated integrals.

The mathematical definitions of various types of integrations over surfaces, volumes etc. does not rule out the possibility that such an integrals might mathematically exist and yet not be computable by doing multiple integration. (As a possible example, we can define what it means to integrate a function over a set in the plane, but the set might be composed of an infinite number of disconnected parts.)