In this note, we record a simple extension of the first fundamental theorem of calculus. Let us recall the first and second fundamental theorems of calculus:
It is well-known that Theorem 1 implies Theorem 2. Indeed, assuming Theorem 1, and for in the assumption of Theorem 2, by letting and using Theorem 1 to differentiate the difference , we can easily get Theorem 2.
Here is the inclusion of in .
It is therefore natural to ask if an extension of Theorem 1 exists in higher dimension. I will illustrate one such extension.
For simplicity let’s assume is continuous and is , where is a bounded open set in and is its closure. Then we have
Theorem 3 If , then
where is the -dimensional Hausdorff measure.
Apply this formula to , then we have
Differentiating both sides with respect to , and applying the first fundamental theorem of calculus (Theorem 1), we can get the result.
Problem: Find the most general settings for this result to hold. Do we have other non-trivial generalization?