Concept

Regular local ring

Related lectures (32)

Green Theorem: Analyzing Potential Derivatives

Explores potential derivatives, Green theorem, simple curves, and adherence in open domains.

Explores the concept of variety defined as the closure of VCA and its applications in algebraic geometry.

Covers the proof of theorem 4.2 on multiplicities and the special structure of local rings at a simple point of a plane.

Regular Functions on Quasi-Affine Varieties

Explores regular functions on quasi-affine varieties, defining morphisms, local rings, and rational functions.

Dimension theory of rings

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Harmonic Functions: Properties and Mean

Covers the properties of harmonic functions, focusing on the mean value property and the maximum principle.

Green's Theorem in 2D: Applications

Explores the applications of Green's Theorem in 2D, emphasizing the importance of regular domains for successful integration.

Projective Plane Curves

Introduces projective plane curves, degrees, components, multiplicities, intersection numbers, tangents, and multiple points, culminating in the statement of Bézout's theorem and its consequences.

Heat Equation: Diffusion

Covers the heat equation for diffusion and conservation of thermal energy.

Divergence Theorem: Green Identities in R²

Explores the divergence theorem and corollaries related to Green identities in the plane, demonstrating their application through examples.

Green Theorem: Proof of Corollary 2

Explains the proof of Corollary 2 of the Green Theorem in a regular domain.

Homogeneous Ideals and Regular Rings

Explores homogeneous ideals, regular rings, N-graded rings, and closed sets in projective geometry.

Fubini Theorem: Integrability and Order of Integration

Explores the Fubini theorem for integrability in two variables and emphasizes the significance of the order of integration.

Green Theorem: Corollary 3 Proof

Demonstrates the relationship between the gradient and the divergence of a scalar field.

Surface Integrals, Divergence Theorem

Covers surface integrals, divergence theorem, and regular domains in 2D and 3D.

Regularity and Noetherian Local Rings

Covers the concepts of regularity and Noetherian local rings in algebraic structures.

Lie Algebras: Introduction and Structure

Introduces Lie algebras, vector spaces with a special bracket operation.

Green's Theorem: Boundary and Normal Vectors

Explores boundaries, normal vectors, and Green's theorem application in transforming integrals.

Volume Calculation in R^3

Covers the calculation of volumes of subsets in R^3 using double integrals.

Local Noetherian Rings

Covers local Noetherian rings, integrally closed domains, discrete valuations, and fraction fields.