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

Lecture

Second Fundamental Tensor

In course

MATH-123(b): Geometry

The course provides an introduction to the study of curves and surfaces in Euclidean spaces. We will learn how we can apply ideas from differential and integral calculus and linear algebra in order to "measure shapes". ...

Description

This lecture covers the concept of the second fundamental tensor, which is a 2x2 symmetric matrix representing the normal unit of a surface. The instructor introduces the tensor as a bilinear area form and explains its role in calculating normal curvature. The lecture also delves into the definition of the tensor, its derivatives, and the dot product in the context of metric tensors.

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

Instructor

Dimitri Stelio Wyss

Official source

Ontological neighbourhood

Mathematics

Geometry: Differential geometry

Algebra: Linear algebra

Related concepts (32)

Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold M (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point p of M is a bilinear form defined on the tangent space at p (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on M consists of a metric tensor at each point p of M that varies smoothly with p.

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then for all indices and Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.

Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if denotes the entry in the -th row and -th column, then the skew-symmetric condition is equivalent to The matrix is skew-symmetric because Throughout, we assume that all matrix entries belong to a field whose characteristic is not equal to 2.

Curvature

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point.

Definite matrix

In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the transpose of . More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of Positive semi-definite matrices are defined similarly, except that the scalars and are required to be positive or zero (that is, nonnegative).

Related lectures (31)

Weingarten Application of Regular Surfaces

Covers the application of the Weingarten map on regular surfaces and the shape operator.

Intersection of Manifolds in Exercise 8.2

Covers the intersection of manifolds in exercise 8.2 and fiscal year 8.8.

Objective function, Differentiability, The second order MOOC: Optimization: principles and algorithms - Unconstrained nonlinear optimization

Explores the role of first and second derivatives in function curvature and convexity.

Covariant Derivatives and Christoffel Symbols

Covers accelerated and inertial coordinate systems, Jacobian, volume elements, covariant derivatives, Christoffel symbols, Lorentz case, and metric tensor properties.

Implicit Surface Parametrizations