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 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
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Category Archives: Calculus
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 singlevalued square root for complex numbers, or … Continue reading
Posted in Analysis, Calculus, Complex analysis, Geometry
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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
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
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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
Posted in Calculus
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Weighted HsiungMinkowski 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
Posted in Calculus, General Relativity, Geometry, Inequalities
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The discrete GaussBonnet theorem
This is a slight extension of my previous note on discrete GaussBonnet theorem. As mentioned in that note, this is a generalization of the wellknown fact that the sum of the exterior angles of a polygon is always , which … Continue reading
Posted in Calculus, Combinatorics, Discrete Mathematics, Geometry, Topology
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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
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