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

Lecture

Green Theorems: Gauss Application

In course

MATH-201: Analysis III

Calcul différentiel et intégral. Eléments d'analyse complexe. ...

Description

This lecture covers the Green and Gauss theorems, focusing on their applications in vector calculus. The theorems are presented in a rigorous mathematical framework, with detailed explanations and examples provided throughout the lecture.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Marco Picasso

Joachim Stubbe

Official source

Related concepts (28)

Divergence theorem

In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, is a theorem which relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed. More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the "flux" through the surface, is equal to the volume integral of the divergence over the region inside the surface.

Green's theorem

In vector calculus, Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. It is the two-dimensional special case of Stokes' theorem. Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then where the path of integration along C is anticlockwise.

Vector calculus

Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations.

Vector calculus identities

The following are important identities involving derivatives and integrals in vector calculus. Gradient For a function in three-dimensional Cartesian coordinate variables, the gradient is the vector field: where i, j, k are the standard unit vectors for the x, y, z-axes. More generally, for a function of n variables , also called a scalar field, the gradient is the vector field: where are orthogonal unit vectors in arbitrary directions. As the name implies, the gradient is proportional to and points in the direction of the function's most rapid (positive) change.

Stokes' theorem

Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.

Related lectures (28)

Warm-up for EPFL: Physics

Offers a warm-up for EPFL students, covering key concepts in mathematics and physics through thematic videos and practical exercises.

Introduction: Mathematical background

Covers basics of modeling for micro/nano devices and emphasizes the importance of computational thinking in understanding and improving device design.

Green's Theorem: Demonstration and Applications

Covers the demonstration and applications of Green's Theorem in vector fields.

Duplication of the Cube

Delves into the historical challenge of duplicating the cube, exploring construction methods, misattributions, and geometric concepts.

Vector Calculus Theorems Log in to Mediaspace to watch this video