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:

Theorem 1 (First fundamental theorem of calculus)If is continuous, then for

Theorem 2 (Second fundamental theorem of calculus)If is , then

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.

It is also well-known that Theorem 2 can be generalized to many situations (usually called Stokes theorem or divergence theorem). E.g. for a smooth form on a compact manifold with boundary , we have

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 3If , then

where is the -dimensional Hausdorff measure.

When , and , we can recover Theorem 1.

**Proof**: We have the coarea formula (which generalizes the Fubini’s theorem)

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?