Tensors Definition and 376 Threads

Is there ever an instance in differential geometry where two different metric tensors describing two completely different spaces manifolds can be used together in one meaningful equation or relation?

Hello, I was pondering on the following: a vector is a specific entity whose existence is independent of the coordinate system used to describe it. To start, I guess I need to state that we are describing the vector from the same reference frame using different coordinate systems (Cartesian...

Is there a purpose of using covariant or contravariant tensors other than convenience or ease in a particular coordinate system? Is it possible to just use one and stick to one? Also what is the meaning of mixed components used in physics , is there a physical significance in choosing one over...

i can't find on internet why higher spin tensors are totally simetric. know this anyone ? I think that is connected to spin matrices.

I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.For the first one I get all my directional derivatives (0, 0): (0)i + (0)j (0, 1): (0)i + 2j (1, 0): 2i + (0)j (1, 1): 2i + 2j Then I square them (FOIL): (0, 0): (0)i...

Hello, I am an undergraduate who has taken basic linear algebra and ODE. As for physics, I have taken an online edX quantum mechanics course. I am looking at studying some of the necessary math and physics needed for QFT and particle physics. It looks like I need tensors and group theory...

Hello, I am an undergrad currently trying to understand General Relativity. I am reading Sean Carroll's Spacetime and Geometry and I understand the physics (to a certain degree) but I am having trouble understanding the notation used as well as the ideas for tensors, dual vectors and the...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...Proposition 12.2 and its proof read as...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Theorem 12.22 on page 276 ... ...The relevant text reads as follows: In the above...

I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Appendix A : Introduction to Tensors ...I need help to understand some statements/equations by Fortney concerning rank one tensors ... Those remarks...

Homework Statement [/B] I am trying to get the C-G Decomposition for 6 ⊗ 3. 2. Homework Equations Neglecting coefficients a tensor can be decomposed into a symmetric part and an antisymmetric part. For the 6 ⊗ 3 = (2,0) ⊗ (1,0) this is: Tij ⊗ Tk = Qijk = (Q{ij}k + Q{ji}k) + (Q[ij]k +...

Hello, I've just been slightly unsure of something and would like to get secondary confirmation as I've just begun a book on tensor analysis. I would also preface this by saying my linear algebra is somewhat rusty. Suppose you have the inertia tensor in some unprimed coordinate system such that...

Homework Statement Solve this, $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}$$ where q is a constant vector. Homework EquationsThe Attempt at a Solution $$\frac{\partial}{\partial x^{\nu}}\frac{3}{(q.x)^3}=3\frac{\partial(q.x)^{-3}}{\partial (q.x)}*\frac{\partial (q.x)}{\partial x^{\nu}}...

Homework Statement Consider the equation for the friction force Ff = m FN. is it possible to express the friction force as a tensor? If so, what rank tensor is it, and what are the ranks of the tensor m and the normal force FN? Homework Equations Ff = mFNThe Attempt at a Solution [/B] So I...

Given some metric, what is an example where the length of a vector is not preserved?

In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...

Is (Tii)2 equivalent to (∑i = 1nTii)2? That is, when you encounter parentheses with Einstein summation, you perform the summation first and then apply any mathematical operations indicated by the parentheses? The solutions manual gives a solution to a problem I've been working out seems to...

Hello PF, I have a question about comparing tensors at different points. Carroll says, “there is no natural way to uniquely move a vector from one tangent space to another; we can always parallel-transport it, but the result depends on the path, and there is no natural choice of which path to...

When dealing with any tensor quantity, when making a coordinate transformation, we should put a bar (or whatever symbol) over the functions or over the indices? For exemple, should the metric coefficients ##g_{\mu \nu}## be written in another coord sys as ##\bar g_{\mu \nu}## or as ##g_{\bar \mu...

Homework Statement I am following a textbook "Seismic Wave Propagation in Stratified Media" by Kennet, I was greeted by the fact that he decided to use cylindrical coordinates to compute the Stress and Strain tensor, so given these two relations, that I believed to be constitutive given an...

Are these two subjects closely related? It seems a tensor can be invariant when viewed from any **co ordinate system and The Lorentz Transformation seems to allow 2 moving co ordinate frames to agree on a space time intervals. Is there some deep connection going on? **=moving frames of...

In one General Relativity paper, the author states the following (we can assume tensor in question are tensors in a vector space ##V##, i.e., they are elements of some tensor power of ##V##) To discuss general properties of tensor symmetries, we shall use the representation theory of the...

I have not really finished studying linear algebra, I have to admit. The furthest I have gotten to is manipulating matrices a little bit (although I have used this in differential equations to calculate a Wronskian to see if two equations are linear independent, but again, a determinant is...

Homework Statement 1. Explain the difference between a covariant tensor and a contravariant tensor, using the metric tensor as an example. 2. Explain how the components of a general covariant tensor may be converted into those of the equivalent contravariant tensor, and vice versa. Homework...

I am quite new to tensors, with my knowledge based on Daniel Fleisch’s “Student’s guide to vectors and tensors” and Neuenschwander’s “Tensor calculus for physics”. I had the following questions: 1. What are the higher rank tensors with physical meaning attached to them? I know tensors up to...

Homework Statement We've been told there's this operation called 'contraction' where if you have a superscript and a subscript that are the same they cancel. I don't understand how that works, partly in the sense that we haven't got round to what the superscripts and subscripts actually mean...

Homework Statement Write out this covariant derivative in terms of partial derivatives and Christoffel symbols: ##\nabla_{\mu} S^{\nu}_{\nu \rho}## Homework EquationsThe Attempt at a Solution I think you can contract that so it reads ##\nabla_{\mu} S_{\rho}##, in which case the solution...

I need help with this problem. This is not a homework assignment, so please don’t send it over to the homework forum. It involves mechanical engineering dynamics that probably are more subtle and advanced then first year mechanical engineering dynamics. It might involve tensor analysis...

Hello there, Recently I encountered a type of covariant derivative problem that I never before encountered: $$ \nabla_\mu (k^\sigma \partial_\sigma l_\nu) $$ My goal: to evaluate this term According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) =...

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...

Hello! I am reading about tensors and I am a bit confused about rank-2 tensors. From what I understand they can be represented by a matrix. However I am not sure I understand the difference between (2,0), (0,2) and (1,1) tensors. I understand that they act on different objects (vectors or one...

I need a good book on tensors, so that I can understand and get good hold of the topic. Can anyone recommend me a good book, like one used in undergraduate level?

In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...

Hello! I am not really sure I understand the idea of tensors and the difference between them and normal matrices, for example (for rank 2 tensors). Can someone explain this to me, or give me a good resource for this? I don't want a complete introduction to GR math, I just want to understand the...

Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...

Hi everyone, I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering : ## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu## My...

Homework Statement I have several problems that ask me to prove that some quantity "transforms like a tensor" For example: "Suppose that for each choice of contravariant vector (a vector) A^nu(x), the quantities B_mu(x) are defined at teach point through a linear relationship of the form...

Looking at relativistic transformations and suddenly we have this transformation matrix with an upper and lower index. See below: A bit of googling tells me the upper index means a co-ordinate. However I'm not sure what the lower index is. Overall I have no idea what makes it so special, or...

Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...

Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...

Hey so probably a really simple question, but I'm stumped. How do you simplify: ν∇⋅(ρν), where ν is a vector ∇ is the "del operator" ⋅ indicates a dot product ρ is a constant. I want to say to do the dyadic product of v and ∇, but then you would get (v_x)*(d/dx) + ... which would be...

Hey guys, Can you please refer some good books to refer to in studying relativistic Electrodynamics (introductory parts), covering the Maxwell's equations in tensor form the L-W potentials and other aspects. FYI am just a beginner in relativistic Electrodynamics. Thanks for the help.

Question 1 - I know a tensor is not a matrix. But the values of each component of a tensor of the form Aμ1μ2 can be arranged in exactly the same way as in a usual 2-dimensional matrix. I was wondering if it would be possible to represent a Aμ1μ2μ3 tensor by a 3-dimensional matrix, and likewise...

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...

I do not get the conceptual difference between Riemann and Ricci tensors. It's obvious for me that Riemann have more information that Ricci, but what information? The Riemann tensor contains all the informations about your space. Riemann tensor appears when you compare the change of the sabe...

Hello again, I would like to know what your opinions about this book. As I have figured out, there are a lot of great GR books out there, but this very rarely gets any mention in forums like this. Why is this? Its got pretty good reviews at amazon and goodreads. Thanks in advance!

I am new to elastic theory. I have a question about elasticity. We assume we have a body with no internal forces. Surface forces are applied on the border. Can we leave the elastic domain (reach the yield surface) in an interior point without leaving the elastic domain on the boundary? If no...

Suppose that in the tensor component ##T^a_b ## the upper index is the ## \bf{3}## component and the lower index is the ##\bf{\bar{3}} ## component. To be concrete, consider the decomposition u^iv_j= \left( u^iv_j-\frac{1}{3}\delta^i_j u^kv_k \right) +\frac{1}{3}\delta^i_j u^kv_k which...

Homework Statement I am meant to show that the following equation is manifestly Lorentz invariant: $$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$ Homework Equations I am given that ##F^{\mu\nu}## is a tensor of rank two. The Attempt at a Solution I was thinking about doing a Lorents...

I'm having a bit of trouble understanding the nature of tensors (which is pretty central to the gen rel course I'm currently taking). I understand that the order (or rank) of a tensor is the dimensionality of the array required to describe it's components, i.e. a 0 rank tensor is a scalar, a 1...