tensor analysis

1. I Transformation rule of contravariant vector

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...
2. Gauge invariance in GR perturbation theory

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...
3. A Normal to coordinate curve

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...
4. I Velocity is a vector in Newtonian mechanics

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 $V$...
5. Finding the inverse metric tensor from a given line element

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...
6. Dust in special relativity - conservation of particle number

1. 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...
7. I Transformation of covariant vector components

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...
8. Calculus Vector Analysis and Cartesian Tensors by Bourne and Kendall

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...
9. I Contraction of mixed tensor

Is that true in general and why: $$A^{mn}_{.~.~lm}=A^{nm}_{.~.~ml}$$
10. Vector Calculus - Tensor Identity Problem

1. Homework Statement 2. Homework Equations 3. 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...
11. Geometry Tensors and Manifolds by Wasserman

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...
12. Differential Geometry book with tensor calculus

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
13. I Exploring Curvature

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 $D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]$...
14. Riemann tensor given the space/metric

1. 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 2. Homework Equations Given the metric and expanding it $~~~g_{τμ} = η_{τμ} + B_{τμ,λσ}x^λx^σ + ...$ We...

18. Metric for the construction of Mercator map

1. 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 $Ω$ 2. Homework...
19. Prove the transformation is scalar

1. 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. 2. Homework Equations 3. The Attempt at a Solution 1.) Rkh = ∂x'h/∂xk By coordinate transformation, x'h = Rkh xk...
20. Construction of an affine tensor of rank 4

1. 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. 2. Homework Equations 3. The...
21. Spin coefficient transformation for null rotation with $l$ fixed

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
22. Transformation rule for product of 3rd, 2nd order tensors

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. 2. Homework Equations : Transformation rule for 3rd order tensors: Z'ijk =...
23. Deriving component form of Riemann tensor in general frame

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...
24. Classical I need a fluid mechanics textbook

Hello, everyone, plz forgive me for my poor English. I'm a sophomore, major in Astronomy. I've finished Hassani's book(Mathematical Physics). And I've learnt Real variable function and functional analysis （I do not know what exact name of this course） I'd like to buy a textbook with following...