What is Tensor calculus: Definition and 100 Discussions
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.
Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.
Homework Statement: The order of indices of the Christoffel symbol of the 1st kind seems to vary from source to source. Is there a preference, and if so why?
Relevant Equations: Christoffel symbol of the 1st kind.
The 1st definition of the Christoffel symbol of the 1st kind I came across was...
Let's say I have the following situation
$$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$
Would I be able to commute the integral and the partial derivative? If so, why is that? In the same line of thought, in the situation I'm able to commute, would the result of...
(I) Using the relevant equation I find this to be ## \frac{e^{x}}{2} ##.
(II) Using the relation for the Ricci tensor, I find that the only non-zero components are...
We know that a metric tensor raises or lowers the indices of a tensor, for e.g. a Levi-Civita tensor. If we are in ##4D## spacetime, then
\begin{align}
g_{mn}\epsilon^{npqr}=\epsilon_{m}{}^{pqr}
\end{align}
where ##g_{mn}## is the metric and ##\epsilon^{npqr}## is the Levi-Civita tensor.
The...
Having two null vectors with $$n^{a} l_{a}=-1, \\ g_{ab}=-(l_{a}n_{b}+n_{a}l_{b}),\\ n^{a}\nabla_{a}n^{b}=0$$ gives $$\nabla_{a}n_{b}=\kappa n_{a}n_{b},\\ \nabla_{a}n^{a}=0,\\ \nabla_{a}l_{b}=-\kappa n_{a}l_{b},\\ \nabla_{a}l^{a}=\kappa$$.
How to show that under the variation of the null...
Hello,
I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
This is probably a stupid question but,
## \frac{d\partial_p}{d\partial_c}=\delta^p_c ##
For the notation of a 4D integral it is ##d^4x=dx^{\nu}##, so if I consider a total derivative:
##\int\limits^{x_f}_{x_i} \partial_{\mu} (\phi) d^4 x = \phi \mid^{x_f}_{x_i} ##
why is there no...
Why do the Cauchy Stress Tensor & the Energy Momentum Tensor have the same SI units? Shouldn't adding time as a dimension changes the Energy Momentum Tensor's units?
Did Einstein start with the Cauchy Tensor when he started working on the right hand side of the field equations of GR?
If so, What...
It's possible that this may be a better fit for the Differential Geometry forum (in which case, please do let me know). However, I'm curious to know whether anyone is aware of any standard naming convention for the two principal invariants of the Weyl tensor. For the Riemann tensor, the names of...
Considering the FLWR metric in cartesian coordinates:
##ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)##
With ##a(t)=t##, the trace of the extrinsic curvature tensor is ##-3t##. But why is it negative if it's describing an expanding universe, not a contracting one?
In a general coordinate system ##\{x^1,..., x^n\}##, the Covariant Gradient of a scalar field ##f:\mathbb{R}^n \rightarrow \mathbb{R}## is given by (using Einstein's notation)
##
\nabla f=\frac{\partial f}{\partial x^{i}} g^{i j} \mathbf{e}_{j}
##
I'm trying to prove that this covariant...
Question ##1##.
Consider the following identity
\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\epsilon_{i}^{\phantom{i}lm}=h^{jl}h^{m}_{\phantom{m}k}-h^{jm}h^{l}_{\phantom{l}k}
\end{equation}
which we know holds in flat space. Does this identity still hold in curved space? and if so, how...
1. The laplacian is defined such that
$$ \vec{\nabla} \cdot \vec{\nabla} V = \nabla_i \nabla^i V = \frac{1}{\sqrt{Z}} \frac{\partial}{\partial Z^{i}} \left(\sqrt{Z} Z^{ij} \frac{\partial V}{\partial Z^{j}}\right)$$
(##Z## is the determinant of the metric tensor, ##Z_i## is a generalized...
I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues.
Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3...
I need to vary w.r.t ##a_{\alpha \beta} ##
##\frac{\partial L}{\partial_{\mu}(\partial_{\mu}{a_{\alpha\beta}})}-\frac{\partial L}{\partial {a_{\alpha \beta}}}## (1)
I am looking at varying the term in the Lagrangian of ##\frac{1}{3}A^{\mu} \partial_{\mu}\Phi ##
where ##A^{\beta}=\partial_k...
I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it.
Tensors. As...
Hi everyone,
studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93)
where $e_r = cos(\theta)e_1 + \sin(\theta)e_2$ and $e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$
How is the...
Assuming Einstein summation convention, suppose $$R^2=\eta_{\mu\nu}x^{\mu}x^{\nu}$$
I was able to show that $$\partial_{\mu}R=\frac{\eta_{\mu\nu} x^{\nu}}{R}$$ by explicitly doing the covariant component of the four-gradient and using the kronecker tensor.
However, how do I use the equation...
Hello.Questions: How tensor operations are done?Like addition, contraction,tensor product, lowering and raising indices. Why do we need lower and upper indices if we want and not only lower? Is a tensor a multilinear mapping?Or a generalisation of a vector and a matrix? Could a tensor be...
I've already made a post about this topic here, but I realized that I didn't understand the explanation on that post. in Chapter 7 of Rindler's book on relativity, in section about electromagnetic field tensor, he states that
_and introducing a factor 1/c for later convenience, we can ‘guess’...
Studying Einstein's original Die Grundlage der allgemeinen Relativitätstheorie, published in 1916's Annalen Der Physik, I came across some equations which I couldn't verify after doing the computations hinted at.
The first are equations 47b) regarding the gravity contribution to the...
I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.
What people usually do is
take the covariant derivative of the covector acting on a vector, the result being a scalar
Invoke a product rule to...
At the end of a long proof I came across something in tensor calculus that seems too good to be true. And if something seems too good to be true ...
The something is that a second order partial derivative vanishes if one of the parts in the denominator is in the same reference frame as the...
Depending on the source, I'll often see EFE written as either covariantly:
$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8 \pi GT_{\mu\nu}$$
or contravariantly
$$R^{\alpha\beta} - \frac{1}{2}Rg^{\alpha\beta} = 8 \pi GT^{\alpha\beta}$$
Physically, historically, and/or pragmatically, is there a...
I've just learned about the covariant derivatives (##\nabla_i## and ##\delta/\delta t##) and I have a doubt.
We should be able to say that $$
J^i = \frac{\delta A^i}{\delta t}
= \frac{\delta^2 V^i}{\delta^2 t}
= \frac{\delta^3 Z^i}{\delta^3 t}
$$ where ##J## is the jolt. This...
I hope you can understand my notation. The Christoffel symbol can be defined through the relation$$
\frac{\partial \pmb{Z}_i} {\partial Z^k} = \Gamma_{ik}^j \pmb{Z}_j
$$ I can solve for the Christoffel symbol this way: $$
\frac{\partial \pmb{Z}_i} {\partial Z^k} \cdot \pmb{Z}^m = \Gamma_{ik}^j...
Greg Bernhardt submitted a new PF Insights post
The 10 Commandments of Index Expressions and Tensor Calculus
Continue reading the Original PF Insights Post.
I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface?
I know how to find equation of normal to a surface. It is given by:
$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
So, I've been studying some tensor calculus for general theory of relativity, and I was reading d'Inverno's book, so out of all exercises in this area(which I all solved), this 6.30. exercise is causing quite some problems, so far. Moreover, I couldn't find anything relevant on the internet that...
Homework Statement
Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid.
Homework Equations
$$\partial _\mu {T^{\mu \nu}} =...
1) Firstly, in the context of a dot product with Einstein notation :
$$\text{d}(\vec{V}\cdot\vec{n} )=\text{d}(v_{i}\dfrac{\text{d}y^{i}}{\text{d}s})$$
with ##\vec{n}## representing the cosine directions vectors, ##v_{i}## the covariant components of ##\vec{V}## vector, ##y^{i}## the...
I would like to know where one may operate with tensor quantities in quantum field theory: Minkowski tensors, spinors, effective lagrangians (for example sigma models or models with four quark interaction), gamma matrices, Grassmann algebra, Lie algebra, fermion determinants and et cetera.
I...
I am using from the following Euler equations :
$$\dfrac{\partial f}{\partial u^{i}}-\dfrac{\text{d}}{\text{d}s}\bigg(\dfrac{\partial f}{\partial u'^{i}}\bigg) =0$$
with function ##f## is equal to :
$$f=g_{ij}\dfrac{\text{d}u^{i}}{\text{d}s}\dfrac{\text{d}u^{j}}{\text{d}s}$$
and we have...
Hi everyone!I'm trying to obtain the natural and dual basis of a circular paraboloid parametrized by:
$$x = \sqrt U cos(V)$$
$$y = \sqrt U sen(V)$$
$$z = U$$
with the inverse relationship:
$$V = \arctan \frac{y}{x}$$
$$U = z$$
The natural basis is:
$$e_U = \frac{\partial \overrightarrow{r}}...
I am currently coding a small application that reproduces the transport of a vector along a geodesic on a 2D sphere.
Here's a capture of this application :
You can see as pink vectors the vectors of curvilinear coordinates and in cyan the transported vector.
The transport of vector along...
I'm going through a basic introduction to tensors, specifically https://web2.ph.utexas.edu/~jcfeng/notes/Tensors_Poor_Man.pdf and I'm confused by the author when he defines vectors as directional derivatives at the bottom of page 3.
He defines a simple example in which
ƒ(x^j) = x^1
and then...
Homework Statement
If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2.
Homework Equations
N/A
The Attempt at a Solution
I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...
Homework Statement
Homework Equations
The Attempt at a Solution
I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus.
This is my attempt at the cross product and...
Hi,
I'm looking for a modern, colourful, illustrative introductory textbook to work through on tensor calculus/continuum mechanics. I'd like one with lots of physical examples, exercises, summaries, etc. I'd like an emphasis on engineering.
Something in the mould of Frank White's Fluid...
I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors.
I looked into the Wikipedia page...
Suppose we are given this definition of the wedge product for two one-forms in the component notation:
$$(A \wedge B)_{\mu\nu}=2A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$$
Now how can we show the switch from tensor products to wedge product below...
The geodesic general condition, i.e. for a non affine parameter, is that the directional covariant derivative is an operator which scales the tangent vector:
$$\zeta^{\mu}\nabla_{\mu}\zeta_{\nu}=\eta(\alpha)\zeta_{\nu}$$
I have three related questions.
When $$\alpha$$ is an affine parameter...
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