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...
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...
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...
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...
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...
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?
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...
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...
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 =...
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...
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...
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...
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...
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?
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...
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...
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?
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...
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...
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...
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 =...
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...
First by "this derivation" I'm referring to an online tutorial: http://farside.ph.utexas.edu/teaching/336L/Fluidhtml/node9.html
It's said in the above tutorial that the ##i-th## component of the total torque acting on a fluid element is
##\tau_i = \int_V \epsilon_{ijk} \cdot x_{j} \cdot F_{k}...
1. Homework Statement
Hi I am reviewing the following document on tensor:
https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf
2. Homework Equations
In the middle of page 27, the author says:
Now, using the covariant representation, the expression $$\vec V=\vec...
In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...
1. Homework Statement
I was working through my text on deriving the tensor for Angular momentum of the sums of elements of a rigid body, I follow it all except for one step. Here is a great page which shows the derivation nicely - http://www.kwon3d.com/theory/moi/iten.html
I follow clearly to...
Hello all,
I have a homework question that I am almost 100% sure that I solved, so I do not believe that this post should go into the "Homework Questions" section. This thread does not have to do with the answer to that homework question anyways, but rather a curiosity about whether or not this...
What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.
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...