Existence of partitions of unity
Existence of partitions of unity

The effective mass tensor in the General Relativity
The effective mass tensor in the General Relativity

... where: ρ*µν = (c2/G)Rµν we will call the effective mass density tensor, ρ = (c2/G)R is the mass density. The simple two-sided multiplication changed the physical sense of the equation (8): in (8) the distribution of matter and energy in the space-time determines his curvature directly and unambiguou ...
ENE 429 Antenna and Transmission Lines
ENE 429 Antenna and Transmission Lines

Rn a vector space over R (or C) with canonical basis {e 1, ...,en
Rn a vector space over R (or C) with canonical basis {e 1, ...,en

Math 217: §2.2 Linear Transformations and Geometry Professor
Math 217: §2.2 Linear Transformations and Geometry Professor

... 1. If we know the values of a linear transformation T : Rn → Rd on each ~ei , do we know the value for any ~x ∈ Rn ? Why? Discuss with your tablemates. 2. Prove that T (~x) = A~x where A is the d × n matrix formed by the vectors T (~e1 ), . . . T (~en ). This is a crucial idea. Be sure you understan ...
Tracking Shape, space and Measure/Geometry Learning Objectvies
Tracking Shape, space and Measure/Geometry Learning Objectvies

Modification of Coulomb`s law in closed spaces
Modification of Coulomb`s law in closed spaces

2(a) Let R be endowed with standard topology. Show that for all x ε
2(a) Let R be endowed with standard topology. Show that for all x ε

Eddington`s Theory of Gravity and Its Progeny
Eddington`s Theory of Gravity and Its Progeny

x and y - Ninova
x and y - Ninova

Document
Document

Exercise Sheet no. 3 of “Topology”
Exercise Sheet no. 3 of “Topology”

S operator( ). 2) Magnetic field is applied along positive Z axis. Find
S operator( ). 2) Magnetic field is applied along positive Z axis. Find

Handout #5 AN INTRODUCTION TO VECTORS Prof. Moseley
Handout #5 AN INTRODUCTION TO VECTORS Prof. Moseley

The geometry of linear equations
The geometry of linear equations

Lecture 13: October 8 Urysohn`s metrization theorem. Today, I want
Lecture 13: October 8 Urysohn`s metrization theorem. Today, I want

The Polarization Vector
The Polarization Vector

Topology Proceedings - Topology Research Group
Topology Proceedings - Topology Research Group

Lecture notes 3
Lecture notes 3

VECTOR ALGEBRA IMPORTANT POINTS TO REMEMBER A
VECTOR ALGEBRA IMPORTANT POINTS TO REMEMBER A

Extended theories of gravity and fundamental physics: Probing the
Extended theories of gravity and fundamental physics: Probing the

Studies D
Studies D

3318 Homework 7
3318 Homework 7

Geometry : Properties of Circles Symmetry properties: (1) Equal
Geometry : Properties of Circles Symmetry properties: (1) Equal

Topology HW7
Topology HW7

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.