An extension of the First Fundamental Theorem of Calculus

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 ${f:[a, b]\rightarrow \mathbb R}$ is continuous, then for ${x\in (a, b),}$ $\displaystyle \frac{d}{dx}\int_a^x f(t)dt= f(x).$
 Theorem 2 (Second fundamental theorem of calculus) If ${F:[a, b]\rightarrow \mathbb R}$ is ${C^1}$, then $\displaystyle \int_a^b F'(t)dt=F(b)-F(a).$

It is well-known that Theorem 1 implies Theorem 2. Indeed, assuming Theorem 1, and for ${F}$ in the assumption of Theorem 2, by letting ${G(x)= \int_a^x F'(t)dt}$ and using Theorem 1 to differentiate the difference ${G(x)-F(x)}$, 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 ${(n-1)}$ form ${\omega}$ on a compact manifold ${M}$ with boundary ${\partial M}$, we have

$\displaystyle \int_M d\omega = \int_{\partial M}\iota^*\omega.$

Here ${\iota}$ is the inclusion of ${\partial M}$ in ${M}$.

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 ${f:\overline \Omega\rightarrow \mathbb R}$ is continuous and ${h: \overline \Omega\rightarrow \mathbb R}$ is ${C^1}$, where ${\Omega}$ is a bounded open set in ${\mathbb R^n}$ and ${\overline \Omega}$ is its closure. Then we have

 Theorem 3 If ${H_{n-1}(h^{-1}(t))<\infty}$, then $\displaystyle \frac{d}{dt}\int_{\{x\in \Omega: h(x)\le t\}} f(x)|\nabla h(x)|dx= \int_{h^{-1}(t)}f(x)dH_{n-1}(x),$ where ${H_{n-1}}$ is the ${(n-1)}$-dimensional Hausdorff measure.

When ${n=1}$, ${\Omega=(a, b)}$ and ${h(t)=t}$, we can recover Theorem 1.
Proof: We have the coarea formula (which generalizes the Fubini’s theorem)

$\displaystyle \int_\Omega f(x)|\nabla h(x)|dx= \int_{-\infty}^\infty \int_{h^{-1}(t)} f(x)dH_{n-1}(x)dt.$

Apply this formula to ${\Omega_s =\{x\in \Omega: h(x)\le s\}}$, then we have

$\displaystyle \int_{\{x\in \Omega: h(x)\le s\}}f(x)|\nabla h(x)|dx= \int_{-\infty}^s \int_{h^{-1}(t)}f(x)dH_{n-1}(x)dt.$

Differentiating both sides with respect to ${s}$, and applying the first fundamental theorem of calculus (Theorem 1), we can get the result. $\Box$

Problem: Find the most general settings for this result to hold. Do we have other non-trivial generalization?

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