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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