# tensor

1. ### Metric tensor problem

My attempt at $g_{\mu \nu}$ for (2) was \begin{pmatrix} -(1-r^2) & 0 & 0 & 0 \\ 0 &\frac{1}{1-r^2} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2 \sin^2(\theta) \end{pmatrix} and the inverse is the reciprocal of the diagonal elements. For (1) however, I can't even think of how to write the...
2. ### I Deriving tensor transformation laws

Hi, I'm worried I've got a grave misunderstanding. Also, throughout this post, a prime mark (') will indicate the transformed versions of my tensor, coordinates, etc. I'm going to define a tensor. $$T^\mu_\nu \partial_\mu \otimes dx^\nu$$ Now I'd like to investigate how the tensor transforms...
3. ### A Index juggling by example of angular momentum tensor

Lets consider the angular momentum tensor (here $m=1$) \begin{equation} L^{ij} = x^iv^j - x^jv^i \end{equation} and rortational velocity of particle (expressed via angular momentum tensor) \begin{equation} v^j = \omega^{jm}x_m. \end{equation} Then \begin{equation} L^{ij} =...

9. ### Inertia tensor of cone around its apex

Im trying to calculate the principals moments of inertia (Ixx Iyy Izz) for the inertia tensor by triple integration using cylindrical coordinates in MATLAB. % Symbolic variables syms r z theta R h M; % R (Radius) h(height) M(Mass) % Ixx unox = int((z^2+(r*sin(theta))^2)*r,z,r,h); % First...
10. ### A Vec norm in polar coord. differs from norm in cartesian coor

I am really confused about coordinate transformations right now, specifically, from cartesian to polar coordinates. A vector in cartesian coordinates is given by $x=x^i \partial_i$ with $\partial_x, \partial_y \in T_p \mathcal{M}$ of some manifold $\mathcal{M}$ and and $x^i$ being some...
11. ### I Riemann Tensor knowing Christoffel symbols (check my result)

I need to find all the non-zero components of the Riemann Tensor in a two-dimensional geometry knowing that the only two non-zero components of the Christoffel symbols are: \Gamma^x_{xx}=\frac{1}{x} and \Gamma^y_{yy}=\frac{2}{y} knowing that: R^\alpha_{\beta\gamma\delta}=\partial_\gamma...
12. ### I Question in Tensor Calculus

I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field newtonian metric tensor $(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))$. I looked at the solution from a manual and it...
13. ### I What is the magnitude of a tensor?

I know that a vector is a tool to help with quantities that have both a magnitude a direction. At a given point in space, a vector has a particular magnitude and direction and if we take any other direction at the same point we can get a projection of this vector in that direction. Tensor is a...
14. ### Band diagram, conductivity tensor

Hello! Does anyone have an idea of how can I obtain information from a band diagram about the directions along which the system conducts best and worst ? Thank you in advanced! :)
15. ### Crystal Symmetry Problem

Hello guys! I have to solve a problem about crystal symmetry, but I am very lost, so I wonder if anyone could guide me. The problem is the following: Using semiclassical transport theory the conductivity tensor can be defined as: σ(k)=e^2·t·v_a(k)·v_b(k) Where e is the electron charge, t...
16. ### Time Derivative of Rank 2 Tensor Determinant

1. Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: $\frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt})$ 2. Homework Equations $det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn}$ 3...
17. ### I Vectors in Minkowski space and parity

It is known that vectors change them sing under the influence of parity when $(x,z,y)$ change into $(-x,-z,-y)$ $$P: y_{i} \rightarrow -y_{i}$$ where $i=1,2,3$ But what about vectors in Minkowski space? Is it true that $$P: y_{\mu} \rightarrow -y_{\mu}$$ where $\mu=0,1,2,3$. If yes how...
18. ### I Pseudotensors in different dimensions

In this topic https://physics.stackexchange.com/questions/129417/what-is-pseudo-tensor one answer was the next: The action of parity on a tensor or pseudotensor depends on the number of indices it has (i.e. its tensor rank): - Tensors of odd rank (e.g. vectors) reverse sign under parity. -...
19. ### A Moment of Inertia Tensor

(Forgive me if this is in the wrong spot) I understand how tensors transform. I can easily type a rule with the differentials of coordinates, say for strain. I also know that the moment of inertia is a tensor. But I cannot see how it transforms as does the standard rules of covariant...
20. ### I Matrix for transforming vector components under rotation

Say we have a matrix L that maps vector components from an unprimed basis to a rotated primed basis according to the rule x'_{i} = L_{ij} x_{j}. x'_i is the ith component in the primed basis and x_{j} the j th component in the original unprimed basis. Now x'_{i} = \overline{e}'_i. \overline{x} =...

24. ### Insights What Is a Tensor? - Comments

fresh_42 submitted a new PF Insights post What Is a Tensor? Continue reading the Original PF Insights Post.
25. ### A Meaning of Slot-Naming Index Notation (tensor conversion)

I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it, cannot deduce answer from the text. (text is concise, I think it assumes a bit of familiarity with the knowledge) (a) Convert the following expressions and equations into...
26. ### Showing Levi-Civita properties in 4 dimensions

first of all english is not my mother tongue sorry. I want to ask if you can help me with some of the properties of the levi-civita symbol. I am showing that $$\epsilon_{ijkl}\epsilon_{ijmn}=2!(\delta_{km}\delta_{ln}-\delta_{kn}\delta_{lm})$$ so i have this...
27. ### I How do i find the eigenvalues of this tough Hamiltonian?

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...
28. ### I Is Second rank tensor always tensor product of two vectors?

Suppose a second rank tensor $T_{ij}$ is given. Can we always express it as the tensor product of two vectors, i.e., $T_{ij}=A_{i}B_{j}$ ? If so, then I have a few more questions: 1. Are those two vectors $A_i$ and $B_j$ unique? 2. How to find out $A_i$ and $B_j$ 3. As $A_i$ and...
29. ### I Why is stress a tensor

Before I go any further, I do understand the ways that mechanical engineering textbooks explain why stress is a tensor. But all of those explanations seem infused with geometry (which I do NOT mean in a negative way at all); and are demonsrtrations. I am searching for a more concise/abstract...
30. ### Struggling immensely with tensors in multivariable calculus

1. 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. 2. Homework Equations N/A 3. The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't...