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Lecture# Surfaces in Space

Description

This lecture covers various surfaces in space, such as paraboloids of revolution, spheres, hyperboloids, and pseudospheres. It explains the analogy between curves in the plane and surfaces in space, using explicit and implicit equations. The lecture also discusses the intersection of surfaces and the parametric equations of curves in space.

Official source

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In course

Instructor

MATH-189: Mathematics

Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique.

Related concepts (192)

Parametric equation

In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (alternatively spelled as parametrisation) of the object.

Hyperboloid structure

Hyperboloid structures are architectural structures designed using a hyperboloid in one sheet. Often these are tall structures, such as towers, where the hyperboloid geometry's structural strength is used to support an object high above the ground. Hyperboloid geometry is often used for decorative effect as well as structural economy. The first hyperboloid structures were built by Russian engineer Vladimir Shukhov (1853–1939), including the Shukhov Tower in Polibino, Dankovsky District, Lipetsk Oblast, Russia.

Curve

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's Elements: "The [curved] line is [...] the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [...

Differential equation

In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.

Great circle

In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry are the natural analog of straight lines in Euclidean space. For any pair of distinct non-antipodal points on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.

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Harmonic Forms: Main TheoremMATH-410: Riemann surfaces

Explores harmonic forms on Riemann surfaces and the uniqueness of solutions to harmonic equations.

Surface Integrals: Regular ParametrizationMATH-201: Analysis III

Covers surface integrals with a focus on regular parametrization and the importance of understanding the normal vector.

Harmonic Forms and Riemann SurfacesMATH-680: Monstrous moonshine

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.