tensor

1. 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...
2. Covariant Derivatives (1st, 2nd) of a Scalar Field

1. Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it 2. Homework Equations Suppose we have X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi...
3. I Lorentz Transformations in the context of tensor analysis

Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...
4. I Definition of stress-energy tensor

Hello! Why is the stress energy tensor defined as a (2 0) tensor? I understand that it needs 2 one-forms as arguments, but using the metric, can't we bring it to (1 1) or (0 2)? So is there is any physical or mathematical reason why it is defined as (2 0), or it is equally right to define it as...
5. I Tensor and vector notation

Hello. I am confused about the notation for tensors and vectors. From what I saw, for a 4-vector the notation is with upper index. But for a second rank tensor (electromagnetic tensor for example) the notation is also upper index. I attached a screenshot of this. Initially I thought that for...
6. I (2,0) tensor is not a tensor product of two vectors?

Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?
7. A Transforming Spin Matrices (Sx, Sy, Sz) to a Spherical Basis

Say I have {S_{x}=\frac{1}{\sqrt{2}}\left(\begin{array}{ccc} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\\ \end{array}\right)} Right now, this spin operator is in the Cartesian basis. I want to transform it into the spherical basis. Since, {\vec{S}} acts like a vector I think that I only need to...
8. B Resistivity tensor and Magnetoresistance

Hey everyone, I'm currently trying to understand the resistivity and conductivity tensor of a 2D sample. If a current carrying metal bar is placed inside a magnetic field the Hall Effect comes in to play. I tried to search for explanations on how to obtain the resistivity tensor of the metal...
9. Finding the geodesic equation from a given line element

1. Homework Statement We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation 2. Homework Equations Line Element: ds^2 = dq^j g_{jk} dq^k Geodesic Equation: \ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m Christoffel Symbol: \Gamma_{km}^j =...
10. Vector to Tensor properties

1. Homework Statement Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2. 2. Homework Equations A tensor transforms under rotations (R) as a vector: Tij'=RinRjmTnm 3. The Attempt at a Solution I wanted to just create the matrix, but I...
11. Stress tensor for non-newtonian fluid

How does one setup the stress tensor for a non-newtonian fluid? I know that for any fluid the normals should be the pressure and for a power law fluid the shear stress in the direction of flow is related by K(du/dy)^n. Does this mean that all other components are 0 for a symmetric pipe or...
12. I Difference between 'Field' (algebra) and 'Field' (geometry)

I am trying to build up a kind of mind map of the following: Module (eg. vector space) Ring (eg Field) Linear algebra (concerning vectors and vector spaces, from what I understood) Multilinear Algebra (analogously concerning tensors and multi-linear maps) Linear maps & Multilinear maps The...
13. Riemann Tensor

Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...
14. What is the trace of a second rank covariant tensor?

What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another $T^i_j$, the trace is $T^k_k$ but what is the trace for $T_{ij}$ because $T_{kk}$ is not even a tensor?
15. A Opposite "sides" of a surface - Differential Geometry.

How, if at all, would differential geometry differ between the opposite "sides" of the surface in question. Simplest example: suppose you look at vectors etc on the outside of a sphere as opposed to the inside. Or a flat plane? Wouldn't one of the coordinates be essentially a mirror while...
16. Electromagnetic Tensor with (-+++) convention

Hi there, Over the last couple of weeks, I have been learning about the relativistic description of electromagnetism through Leonard Susskind's Theoretical Minimum lectures, and although I have managed to follow it, there are some parts which I am becoming increasingly confused by, not helped...
17. Are 10 dimensions related to the tensor of 3d

I think I've read the the tensor in three dimensions has 10 elements in its matrix(?). Is this related to the 10 dimensions in some forms of string theory?
18. Is the moment of inertia matrix a tensor?

1. Homework Statement Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors. I is the inertia matrix L is the angular momentum \omega is the angular velocity 2. Homework Equations The transformation rule...
19. Demonstrate the matrix represents a 2nd order tensor

1. Homework Statement Demonstrate that matrix $T$ represents a 2nd order tensor $T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}$ 2. Homework Equations To show that something is a tensor, it must transform by $T_{ij}' = L_{il}L_{jm}T_{lm}$. I cannot find a neat general form for...
20. Finding principal axes of electromagnetic stress tensor

1. Homework Statement In a certain system of units the electromagnetic stress tensor is given by M_{ij} = E_iE_j + B_i B_j - \frac12 \delta_{ij} ( E_kE_k + B_kB_k) where E_i and B_i are components of the 1-st order tensors representing the electric and magnetic fields \bar{E} and...
21. 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 =...
22. Metric tensor rank (1,1)

I read in many books the metric tensor is rank (0,2), its inverse is (2,0) and has some property such as $g^{\mu\nu}g_{\nu\sigma}=\delta^\mu_\sigma$ etc. My question is: what does $g^\mu_\nu$ mean?! This tensor really confuses me! At first, I simply thought that...

30. A suggested operational definition of tensors

The two tensor definitions I'm (newly) familiar with, by transformation rules, and as a map from a tensor product space to the reals, don't tell me what a tensor does, and to the best of my knowledge they don't make it apparent. So, I'm looking for an operational definition, and suggesting the...