Category Archives: Calculus

Least squares in a non-ordinary sense

Simple ordinary least squares regression (SOLSR) means the following. Given data , , find a line in represented by that fits the data in the following sense. The loss of each data point to the line is       … Continue reading

Posted in Applied mathematics, Calculus, Optimization, Statistics | Leave a comment

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

Posted in Calculus, Geometry | Leave a comment

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

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

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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