Lecture 6 outline copy
Lecture 6 outline copy

General Relativity: An Informal Primer 1 Introduction
General Relativity: An Informal Primer 1 Introduction

Chapter 1 Vectors
Chapter 1 Vectors

review sheet
review sheet

1300Y Geometry and Topology, Assignment 1 Exercise 1. Let Γ be a
1300Y Geometry and Topology, Assignment 1 Exercise 1. Let Γ be a

Problem Set 1
Problem Set 1

3.1 Properties of vector fields
3.1 Properties of vector fields

8-7 Practice: Vectors
8-7 Practice: Vectors

Document
Document

A LOCALLY COMPACT SEPARABLE METRIC SPACE IS ALMOST
A LOCALLY COMPACT SEPARABLE METRIC SPACE IS ALMOST

Chapter 5
Chapter 5

VECTOR ADDITION
VECTOR ADDITION

1 8. CONSERVATION LAWS The general form of a conservation law
1 8. CONSERVATION LAWS The general form of a conservation law

... stress tensor. Note that the magnetic field transports (or carries) momentum. The quantity p + B 2 / 2 µ0 is the total pressure. The second contribution is called the magnetic pressure; the magnetic field resists compression, just like the fluid pressure. The tensor BB / µ0 is called the hoop stres ...
Ch 3: Motion in 2 and 3-D 3-1 The Displacement Vector DEF
Ch 3: Motion in 2 and 3-D 3-1 The Displacement Vector DEF

Slides
Slides

Geometry review, part I Geometry review I
Geometry review, part I Geometry review I

ECE Lecture 22: Electrostatics – Coulomb`s Law
ECE Lecture 22: Electrostatics – Coulomb`s Law

characterization of curves that lie on a surface in euclidean space
characterization of curves that lie on a surface in euclidean space

21. Metric spaces (continued). Lemma: If d is a metric on X and A
21. Metric spaces (continued). Lemma: If d is a metric on X and A

... f, g : X → R are continuous, then f + g, f − g, f · g are continuous. If g(x) 6= 0 for all x, then f /g is continuous. Defn: Let fn : X → Y be a sequence of functions from the set X to the topological space Y . Then the sequence of functions (fn ) converges to the function f : X → Y if the sequence ...
Poynting Vector and Power Flow in Electromagnetic Fields
Poynting Vector and Power Flow in Electromagnetic Fields

LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

Exercise Sheet no. 1 of “Topology”
Exercise Sheet no. 1 of “Topology”

Document
Document

Section_08_Conservat..
Section_08_Conservat..

PHYS4210 Electromagnetic Theory Spring 2009 Midterm Exam #1
PHYS4210 Electromagnetic Theory Spring 2009 Midterm Exam #1

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of, and angle between, tangent vectors.A metric tensor is called positive-definite if it assigns a positive value to every nonzero vector. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. By integration, the metric tensor allows one to define and compute the length of curves on the manifold. The curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p,q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.With a holonomic basis on the manifold, a metric tensor takes on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.