Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Green's Theorem: Surface Integrals

In course

MATH-203(c): Analysis III

Le cours étudie les concepts fondamentaux de l'analyse vectorielle et l'analyse de Fourier en vue de leur utilisation pour résoudre des problèmes pluridisciplinaires d'ingénierie scientifique. ...

Description

This lecture covers the application of Green's Theorem to surface integrals, focusing on the corollaries and examples related to regular surfaces and coordinate transformations. The instructor explains the concept of exterior and normal surfaces, emphasizing the importance of proper parametrization and coordinate choices. Various formulas and identities are presented to facilitate the calculation of surface integrals, with a detailed analysis of the theoretical background and practical applications.

Instructors (2)

David Valentin Strütt

Sébastien Basterrechea

Official source

Ontological neighbourhood

Mathematics

Analysis: Vector analysis

Related concepts (32)

Parametric surface

A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.

Surface integral

In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate a scalar field (that is, a function of position which returns a scalar as a value) over the surface, or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration.

Implicit surface

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z. The graph of a function is usually described by an equation and is called an explicit representation. The third essential description of a surface is the parametric one: where the x-, y- and z-coordinates of surface points are represented by three functions depending on common parameters .

Green's identities

In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem. This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X ) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ Rd, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable.

Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface.

Related lectures (35)

Differential Forms Integration

Covers the integration of differential forms on smooth manifolds, including the concepts of closed and exact forms.

Surface Integrals: Parameterized Surfaces

Explores surface integrals over parameterized orientable surfaces and their applications in flux and work evaluation.

Understanding Jacobian and Normal Vectors in Surfaces

Clarifies the use of Jacobian, normal vectors in surfaces, and arccos(z) constraints.

Angle Calculation on Regular Surfaces

Covers the calculation of angles between curves on regular surfaces and the concept of curvilinear abscissa.

Curves with Poritsky Property and Liouville Nets