Tensor calculus Definition and 102 Threads

I've seen the terms tensor calculus and tensor analysis both being used - what is the difference?

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

I'm having trouble evaluating the following expression (LATEX): ##\nabla_{i}\nabla_{j}T^{k}= \nabla_{i} \frac{\delta T^{k}}{\delta z^{j}} + \Gamma^{k}_{i m} \frac{\delta T^{m}}{\delta z^{i}} + \Gamma^{k}_{i m} \Gamma^{m}_{i l} T^{l}## What are the next steps to complete the covariant...

Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...

I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...

I'm reading Zee's Gravity book, can anyone help me understand the explanation on this part, I understand everything except the last part, he said to use (I.4.14) so that I could solve for the quantity shown in the image, what does he mean by that and how?

I'm reading A. Zee's GR book and I'm in the section in which he is showing how to transform coordinates to be locally flat in a neighborhood of a point. He said that we can always choose our neighborhood to be locally flat for any space of any dimension D. "Look at how the metric transforms...

Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...

I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...

If you don't like indexes, look away now. I got these terms from a tensor calculus program as part of a the transformed F-P Lagrangian. \begin{align} {g}^{b a}\,{g}^{d e}\,{g}^{f c}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c f}\,{g}^{e d}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c...

At low speeds and assuming pressure ##P=0##, T^{\alpha \beta} = \rho U^\alpha U^\beta g_{\alpha \mu} g_{\gamma \beta} T^{\alpha \beta} = \rho g_{\alpha \mu} g_{\gamma \beta} U^\alpha U^\beta T_{\gamma \mu} = \rho U_\mu U^\beta g_{\gamma \beta} Setting ##\gamma = \mu = 0##: T_{00} = \rho...

What do they mean by contracting ##\mu## with ##\alpha## ?

Homework Statement (a) Show acceleration is perpendicular to velocity (b)Show the following relations (c) Show the continuity equation (d) Show if P = 0 geodesics obey: Homework EquationsThe Attempt at a SolutionPart (a) U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...

Homework Statement (a)Find Christoffel symbols (b) Show the particles are at rest, hence ##t= \tau##. Find the Ricci tensors (c) Find zeroth component of Einstein Tensor Homework EquationsThe Attempt at a Solution Part (a)[/B] Let lagrangian be: -c^2 \left( \frac{dt}{d\tau}\right)^2 +...

Homework Statement [/B] (a) Find christoffel symbols and ricci tensor (b) Find the transformation to the usual flat space form ## g_{\mu v} = diag (-1,1,1,1)##. Homework EquationsThe Attempt at a Solution Part(a) [/B] I have found the metric to be ## g_{tt} = g^{tt} = -1, g_{xt} = g_{tx} =...

Hello Big minds, In the book of Arfken [Math Meth for Physicists] p 134 he defined contravariant tensor...my question is about a_ij he defined them first as cosines of an angle of basis then he suddenly replaced them by differential notation...why is that? cosines are not mention in this...

Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...

Hi all, I'm fairly new to GR, and I'm also somewhat new to tensors as well. I'm looking for some detailed explanation of a tensor, as I want to begin studying GR mathematically. I watched a video that was posted on PF not too long ago that was pretty good. I'm having trouble remembering who it...

Hello everyone: I'm confusing with the construction and application of dyadic green's function. If we are in the ideal resonant system where only certain resonant mode is supported in this space (such as cavity), the Green's function can be constructed by the mode expansion that is: Gij(r,r')...

Homework Statement (a) Find faraday tensor in terms of ##\vec E## and ## \vec B ##. (b) Obtain two of maxwell equations using the field relation. Obtain the other two maxwell equations using 4-potentials. (c) Find top row of stress-energy tensor. Show how the b=0 component relates to j...

Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms. I have, in particular: \displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p}...

I'm a mathematics major and up until now I've taken Calc 1,2,3 (so single + multivariable) a combined course in Elementary Linear Algebra + Differential Equations and PDE's. My school doesn't offer any tensor calculus classes, but I was interested in learning some of it on my own. Do I have...

I have started to learn a bit about Tensor calculus and it all going above my head. May anyone give a brief outline about the topic (preferably theoretical) and the supplementary concepts attached to it.

How do we explains space-time through Riemann Calculus?

In an attempt to solve the mystery of dark energy, I came across problems concerned with the General Relativity. In it, I observed that many of the problems were related with the tensor calculus. I want to know that what importance does tensor calculus hold in GR? Are there any other fields of...

Hello everyone! Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider $$R_{\mu\nu} = 0$$. If I expand the Ricci tensor, I get $$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$. Which, in normal algebra, should...

Hello Everyone, I have read many derivations of Einstein field equations (done one myself), but none of them explain why the constant term should have a $$c^4$$ in the denominator. the 8πG term can be obtained from Poisson's equation, but how does c^4 pop up? Most of the books just derive it...

I already have the solutions emailed to me from a D H Lawden textbook. I have trouble understanding the solution as the solution is not formatted properly, and the answer seems to be a little too advanced for me. I hope that some one can help me understand the problem. 1. Homework Statement...

Hi there. I was dealing with the derivation on continuum mechanics for the conservation of angular momentum. The derivation I was studying uses an arbitrary constant skew tensor ##\Lambda##. It denotes by ##\lambda## its axial vector, so that ##\Lambda=\lambda \times## Then it defines...

Hi guys, I am interested to learn tensor calculus but I can't find a good book that provide rigorous treatment to tensor calculus if anyone could recommend me to one I would be very pleased.

Hello all, After a brief break from attempting to learn tensor calculus, I'm once again back at it. Today, I started reading this: http://web.mit.edu/edbert/GR/gr1.pdf. I got to about page 4 before things stopped making sense, right under equation 3. Question 1: apparently a "one-form" is a...

I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...

Hey everyone, I recently learned that my certified genius weird-uncle-who-lives-at-home (IQ over 200 something, legitimate 'genius') or WULAH for short, passes his spare time by lounging around his place and doing tensor calculus. I've done some calc in 3d in college and I know that's commonly...

Hello, could someone recommend a good book on tensor calculus? I'd like it to be relatively modern (I have an old book) and maybe contain some examples drawn from physics. Chapters on related subjects such as differential forms and calculus of variations would be a plus. Cheers.

Use the metic that Einstein proposed in the first cosmological model based on general relativity. ds2 = -dt2 + (dr2) / (1 - Kr2) + r2(dθ2 + sin2θd\phi2) where K > 0 Show that the stress energy tensor is that of a static, spatially uniform perfect fluid and determine ρ and p in terms of G and...

Hi Let D be an anisotropic tensor. This means especially, that D is traceless. \mathrm{tr}(D) = 0 Apply the representating matrix of D to a basis vector S , get a new vector and multiply this by dot product to your basis vector. Than you got a scalar function. Now integrate this...

Question: Let Aij denote an absolute covariant tensor of order 2. Show that the determinant A = det(Aij ) is an invariant of weight 2 and A is an invariant of weight 1. I have little clue about this question. Would writting down the transformation rule from barred to unbarred 2nd-order tensor...

Hi, the only thing I know is differential and integral calculus for functions with 1 variable and the basics of linear algebra (solving linear systems with matrices, determinants, etc...) . I'm currently learning differential and integral calculus for real functions with multiple real variables...

Homework Statement if BijkAjk is a vector for all symmetric tensors Ajk, (but Bijk is not necessarily a tensor), what are the properties of Bijk under rotations of the basis/coordinate axes? Homework Equations The Attempt at a Solution I am not sure what the question is looking for... though I...

Hello I have huge problems with the following exercise. Please give me some hints. No complete Solutions but a little bit help. Find the differential equations of the paths of test particles in the space-time of which the metric ist \mathrm{d}s^2 = e^{2kx} \left[- \left( \mathrm{d}x^2...

I have a book that I've been reading off and on: https://www.amazon.com/dp/012200681X/?tag=pfamazon01-20 It's enjoyable but the only issue I'm having with it is the notation used, still. The book is definitely good at what it wants to achieve, which is a bridging text to the math for...

Dear Friends I have two questions to do about Tensor Calculus: 1) Is there any program to calculate Christoffel Symbols, Riemann and Ricci Tensors and everything about Tensor Calculus (Free or paid)? 2) When in an exercise anyone asks to use the Euclidean metric or Riemann metric, what...

I'm taking a course on relativity, both special and general. According to my college, I have the required mathematical background (vector analysis, electromagnetics (though I can't recall more than a cursory glance at tensors) etc) to make sense of it. Special relativity I can handle, and I...

the book "Introduction to tensor calculus and continuum mechanics" hi please help me ! i've try to find appendix d of this or solution that excersise anyone can help me ? any link or full version of this book thart have "appendix d" my english languge not good sorry thanks a lot bye

I am studying Schaum's Tensor Calculus by Kay. I am attempting to work through every solved problem (covering up the answers, first) and every supplementary problem. I am not a student. My day job is computational chemistry, so I can only do this in my spare time (whatever that is!). A...

I am trying to teach myself Tensor Calculus from a book. I am stuck. Websites and reading don't help. I need to see how the problems are actually done. Does anyone know of any videos or tutors that can help? Tried everything but no luck. At least if you can see a problem worked in class then you...

Using the Einstein convention, is this about right? (indexes run from 1 to 3): \nabla\bullet(x_{i}a)=div(x_{i}a)=\partial_{j}a_{j}x_{i}=3x_{i}

Homework Statement equation of ellipse may be written ax^2 + 2hxy + by^2 =1 What is the tensor charater of a, h, b with respect to transformation to any Cartesian coordinates(rectangle or oblique) in the plane? Homework Equations none The Attempt at a Solution no idea

Here is a free and downloadable textbook on Tensor Calculus. http://www.math.odu.edu/~jhh/counter2.html Scroll to bottom of page to find pdf files.

Hi, Anyone out there have Lawden's book, "An Introduction to Tensor Calculus, Relativity and Cosmology"? Expression 39.13 (on p 109) and the sentence that precedes it have me stumped. My understanding of what he's saying is that a symmetric quadratic form can always be diagonalized so that...