# Thread: Surface Integral

1. ## Surface Integral

What's the difference between a surface integral with an S written underneath it and the one with two ∫ signs enclosed with a circle?

Is a surface integral always with respect to the change in area (i.e. it is just a special case of integration where we integrate with respect to change in area), and we have named that case a surface integral?

2. ## Re: Surface Integral

When there is a circle around the integral signs that means that $S$ is a closed surface. In fact a surface integral is a special case of integrating what is called a differential form. In the case of a surface integral we are integrating what is known as a 2-form: for a line integral we integrate a 1-form and in the case of an ordinary integral of a function we are integrating a 0-form. Indeed there are higher forms that can be integrated such as in volume integrals (integrating a 3-form). Now to directly answer your question: yes surface integrals always sum differential area elements.