In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a pure number plus a direction, like velocity), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space.
Many mathematical structures called "tensors" are tensor fields. For example, the Riemann curvature tensor is not a tensor, as the name implies, but a tensor field: It is named after Bernhard Riemann, and associates a tensor to each point of a Riemannian manifold, which is a topological space.
Divergence formula
$$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$
If we express it in terms of the components of ##\vec{A}## in unit basis using
$$A^{*j} = \sqrt{g^{jj}} A^{j}$$
, we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}}...
May I recommend math3d.org as a website for making 3D-Graphics for fresh_42's page on List of Online Calculators for Math, Physics, Earth and Other Curiosities.
I also might recommend cloudconvert.com to make .gifs. It is simpler than ezgif.com, mainly because it is not so feature rich.
Here...
Hi,
I would like to ask for a clarification about the difference between a differential k-form and a generic (0,k) tensor field.
Take for instance a (non simple) differential 2-form defined on a 2D differential manifold with coordinates ##\{x^{\mu}\}##. It can be assigned as linear combination...
Let ##\varphi## be some scalar field. In "The Classical Theory of Fields" by Landau it is claimed that
$$
\frac{\partial\varphi}{\partial x_i} = g^{ik} \frac{\partial \varphi}{\partial x^k}
$$
I wanted to prove this identity. Using the chain rule
$$
\frac{\partial}{\partial x_{i}}=\frac{\partial...
I am having trouble to understand what it means by "physically relevant real parameters" and how does it help us to specify a quantum system.
Let say, we have a state of k half spin electrons? My guess is about the local phase of the spin, and this would make it 2^k parameters since each...
I got stuck in this calculation, I can't collect everything in terms of ##dx^{\mu}##.
##x'^{\mu}=\frac{x^{\mu}-x^2a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2x^2}##
##x'^{\mu}=\frac{x^{\mu}-g_{\alpha \beta}x^{\alpha}x^{\beta}a^{\mu}}{1-2a_{\nu}x^{\nu}+a^2g_{\alpha \beta}x^{\alpha}x^{\beta}}##...
I know that a tensor can be seen as a linear function.
I know that the tensor product of three spaces can be seen as a multilinear map satisfying distributivity by addition and associativity in multiplication by a scalar.
##(\nabla\times\vec B) \times \vec B=\nabla \cdot (\vec B\vec B -\frac 1 2B^2\mathcal I)-(\nabla \cdot \vec B)\vec B##
##\mathcal I## is the unit tensor
I kinda know how to do this problem, it is just that I hit a sign problem. If I take the partial derivative of the coordinate transformation with respect to ##x'^\mu##, I get
writing it first in the inverse form, ##x^\alpha = x'^\alpha - \epsilon^\alpha##
##\frac{\partial x^\alpha}{\partial...
I am trying to answer exercise 5 but I am not sure I understand what the hint is implying, differentiate with respect to ##p_\alpha## and ##p_\beta##, I have done this but nothing is clicking. Also, what is the relevance of the hint "the constraint ##p^\alpha p_\alpha = m^2c^2## can be ignored...
Since ##\nu## is contracted, we form the scalar product of the metric and inverse metric,
##g_{\mu\nu}g^{\nu\lambda} = (\vec{e_\mu} \cdot \vec{e_\nu}) \cdot (\vec{e^\nu} \cdot \vec{e^\lambda}) = \vec{e_\mu} \cdot (\vec{e_\nu} \cdot \vec{e^\nu}) \cdot \vec{e^\lambda} = \delta^\lambda_\mu##
I...
In a spherical polar coordinate system if the components of a vector given be (r,θ,φ)=1,2,3 respectively. Then the component of the vector along the x-direction of a cartesian coordinate system is $$rsinθcosφ$$.
But from the transformation of contravariant vector...
I have been following [this video lecture][1] on how to find gauge invariance when studying the perturbation of the metric.
Something is unclear when we try to find fake vs. real perturbation of the metric.
We use an arbitrary small vector field to have the effect of a chart transition map or...
Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface?
I know how to find equation of normal to a surface. It is given by:
$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$
However the answer is given using metric tensor which I am not able to derive. Here is the...
I studied the vector analysis in Arfken and Weber's textbook : Mathematical Methods for Physicists 5th edition.
In this book they give the definition of vectors in N dimensions as the following:
The set of ##N## quantities ##V_{j}## is said to be the components of an N-dimensional vector ##V##...
Defining dS2 as gijdxidxj and
given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gij
Now here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...
Homework Statement
My textbook states:
Since the number of particles of dust is conserved we also have the conservation equation
$$\nabla_\mu (\rho u^\mu)=0$$
Where ##\rho=nm=N/(\mathrm{d}x \cdot \mathrm{d}y \cdot \mathrm{d}z) m## is the mass per infinitesimal volume and ## (u^\mu) ## is...
Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$:
$$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$
And
$$ \textbf{e}^i = \nabla u^{i} $$
In the primed...
I have to do a teaching assistant job on a multivariable calculus class, I have to survey books that can be useful as resources. Has anyone used this book by Bourne and Kendall? I noticed that the treatment of vector analysis seems good and the chapter on Cartesian tensors seem to be a good...
Homework Statement
Homework Equations
The Attempt at a Solution
I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus.
This is my attempt at the cross product and...
I would like to know at what level is the book Tensors and Manifolds by Wasserman is pitched and what are the prerequisites of this book? Given the prerequisites, at what level should it be (please give examples of books)? If anyone has used this book can you please kindly give your comments and...
Hi, there is a book of dg of surfaces that is also about tensor calculus ?
Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus!
Thank you in advance
Here's what I'm watching:
At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate
##D_s D_r T_m - D_r D_s T_m##
And on the last term I'm having some difficulties, the second christoffel symbol.
So we have
##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...
Homework Statement
Given two spaces described by
##ds^2 = (1+u^2)du^2 + (1+4v^2)dv^2 + 2(2v-u)dudv##
##ds^2 = (1+u^2)du^2 + (1+2v^2)dv^2 + 2(2v-u)dudv##
Calculate the Riemann tensor
Homework Equations
Given the metric and expanding it ##~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...##
We have...
Homework Statement
Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)##
Homework Equations
##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ##
##D_vU^ρ = ∂_vU^ρ +...
Homework Statement
Find the locally flat coordinates on the Poincaré half plane.
Problem I.6.4 by A. Zee
Homework Equations
[/B]
Poincaré Metric: ##ds^2 = \frac{dx^2 + dy^2}{y^2}##
The Attempt at a Solution
First, I'm having problems with the explanation in Zee's book. He said that we can...
Homework Statement
Given the line element ##ds^2## in some space, find the transformation relating the coordinates ##x,y ## and ##\bar x, \bar y##.
Homework Equations
##ds^2 = (1 - \frac{y^2}{3}) dx^2 + (1 - \frac{x^2}{3}) dy^2 + \frac{2}{3}xy dxdy##
##ds^2 = (1 + (a\bar x + c\bar y)^2) d\bar...
Homework Statement
The familiar Mercator map of the world is obtained by transforming spherical coordinates θ , ϕ to coordinates x , y given by
##x = \frac{W}{2π} φ,
y = -\frac{W}{2π} log (tan (\frac{Θ}{2}))##
Show that ##ds^2 = Ω^2(x,y) (dx^2 + dy^2)## and find ##Ω##
Homework Equations...
Homework Statement
1.) Prove that the infinitesimal volume element d3x is a scalar
2.) Let Aijk be a totally antisymmetric tensor. Prove that it transforms as a scalar.
Homework Equations
The Attempt at a Solution [/B]
1.) Rkh = ∂x'h/∂xk
By coordinate transformation, x'h = Rkh xk
dx'h =...
Homework Statement
In En the quantities Bij are the components of an affine tensor of rank 2. Construct two affine tensors each of rank 4, with components Cijkl and Dijkl for which
∑k ∑l Cijkl Bkl = Bij + Bji
∑k ∑l Dijkl Bkl = Bij - Bji
are identities.
Homework Equations
The Attempt at a...
In Newmann-Penrose formalism, a Null rotation with ##l## fixed is
$$l^a−>l^a\\
n^a−>n^a+\bar{c}m^a+c\bar{m}^a+c\bar{c}l^a\\
m^a−>m^a+cl^a\\
\bar{m}^a−>\bar{m}^a+\bar{c}l^a$$
Using this transformation, how to prove?
$$π−>π+2\bar{c}ϵ+\bar{c}^2κ+D\bar{c}$$
Ref: 2-Spinors by P.O'Donell, p.no, 65
1. Problem statement:
Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Zijk such that the relation ui = ZijkAjk remains valid after a coordinate rotation.
Homework Equations :
[/B]
Transformation rule for 3rd order tensors: Z'ijk =...
How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection?
I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...
Hello, everyone, please forgive me for my poor English.
I'm a sophomore, major in Astronomy. I've finished Hassani's book(Mathematical Physics). And I've learned Real variable function and functional analysis （I do not know what exact name of this course）
I'd like to buy a textbook with...