Category Archives: Geometry

Elementary geometry, differential geometry, Riemannian geometry, algebraic geometry.

Archimedes’ principle for hyperbolic plane

After writing the previous post, I realized that there is an exact analogue of the Archimedes’ principle for the surface area of the hyperbolic disk inside the hyperbolic plane. Let us fix the notations. Let be the -dimensional Minkowski space … Continue reading

Posted in Calculus, Differential geometry, Geometry | Leave a comment

Archimedes and the area of sphere

I record here a remarkable discovery of Archimedes about the formula for the surface area of a sphere. I think the derivation is elementary enough so that it can be taught in high school. (I think I was not taught … Continue reading

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Euler’s formula e^ix = cos x + i sin x: a geometric approach

Today I mentioned the famous Euler’s formula briefly in my calculus class (when discussing hyperbolic functions, lecture notes here): where is a solution to (usually denoted by “”, but indeed there is no single-valued square root for complex numbers, or … Continue reading

Posted in Analysis, Calculus, Complex analysis, Geometry | Leave a comment

An inequality for functions on the plane

I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also … Continue reading

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Weighted isoperimetric inequalities in warped product manifolds

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading

Posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities | 2 Comments

Faber-Krahn inequality

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading

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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