Category Archives: Calculus

Why is a² + b² ≥ 2ab ?

This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading

Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability | 1 Comment

A remark on the divergence theorem

The divergence theorem states that for a compact domain in with piecewise smooth boundary , then for a smooth vector field on , we have where is the unit outward normal and is the divergence of . In most textbooks, … Continue reading

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Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

1. Motivation and Main Results A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in are round hyperspheres. The embeddedness assumption is essential. For instance, admits immersed tori with constant mean … Continue reading

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The discrete Gauss-Bonnet theorem

This is a slight extension of my previous note on discrete Gauss-Bonnet theorem. As mentioned in that note, this is a generalization of the well-known fact that the sum of the exterior angles of a polygon is always , which … Continue reading

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Why a vector field rotates about its curl?

Let be a smooth vector field on . We define its curl to be the vector field It is often claimed in advanced calculus that measures how fast “rotates”. More specifically, suppose , then locally around , the vector field … Continue reading

Posted in Calculus, Linear Algebra | 1 Comment

A functional inequality on the boundary of static manifolds

1. introduction and statement of results The research in this article is largely motivated by the following result concerning a functional inequality on the boundary of bounded domains in the Euclidean space , proved in [MTX] Corollary 3.1. Theorem 1 … Continue reading

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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 is continuous, then for   Theorem … Continue reading

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