In mathematics, a tensor is an algebraic object that describes a (multilinear) relationship between sets of algebraic objects related to a vector space. Objects that tensors may map between include vectors and scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 - continuing the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others - as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.
A brief explanation of vector and tensor concepts from A Student's Guide to Vectors and Tensors by Dan Fleisch. I found this when I was trying to better understand tensors and how they are used.
TL;DR Summary: Pretty much confused about an advanced elasticity book.Resource recommendation is asked.
My last semester in freshman year of bs physics included a chapter on elasticity,it was not at the advanced level and by advanced level i mean atleast the tensor stuff.Well,I want to read...
Exercise:
Solution:
The result is correct, but I'm unsure about equation from 29 to 30 where right-hand side became just the covariant dual field tensor. I assumed that I could interchange the covariant dual- and normal covariant field tensor, but don't think it's possible since matrices...
New to group theory. I have 3 questions:
1. Tensor decomposition into Tab = T[ab] +T(traceless){ab} + Tr(T{ab}) leads to invariant subspaces. Is this enough to imply these subreps are irreducible?
2. The Symn representations of a group are irreps. Why?
3. What is the connection between...
I simply just wrote down the definition of ##\Gamma'^a_{bc}##, and inserted the transformations of ##g'^{ad}##, ##g'_{dc,b}##, and the like terms. After some rearranging and cancelling out,
$$\Gamma'^a_{bc}=\frac{\partial x'^a}{\partial x^e}\frac{\partial x^f}{\partial x'^b}\frac{\partial...
Hi, I'm struggling with understanding the idea of tensor product and direct sum beyond the very basics. I know that direct sum of 2 vectors basically stacks one on top of another - I don't understand more than this . For tensor product I know that for a product of 2 matrices A and B the tensor...
Dirac's book "General Theory of Relativity" says on p. 2 that a general rank-2 tensor can be written as a sum of outer products: $$ T^{\mu\nu} = A^\mu B^\nu + A'^\mu B'^\nu + A''^\mu B''^\nu + \cdots $$ Importantly, he repeats this on p. 18, in developing the covariant derivative, where he...
I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in confidently interpreting his use of Dirac Notation in Section 1.9 ...
in Section 1.9 we read the following:
I need some help to confidently interpret and proceed with Neuenschwander's notation...
ep_{ijkl} M^{ij} N^{kl} + ep_{ijkl}N^{ij} M^{kl}
The second term can be rewritten with indices swapped
ep_{klij} N^{kl}M^{ij}
Shuffle indices around in epsilon
ep{klij} = ep{ijkl}
Therefore the expression becomes
2ep_{ijkl}M^{ij}N^{kl}
Not zero.
What is wrong here?
How can we (as nicely as possible... i.e. not via characteristic polynomial) calculate the eigenvalues of ##F_{ab} = \partial_a A_b -\partial_b A_a## and ##T_{ab} = F_{ac} {F_b}^c- (1/4) \eta_{ab} F^2 ## and what is their physical meaning?
Hello,
I'm reading Group Theory in a nutshell for physicist by A Zee. When he introduced Dual tensors (pp 192), he made a claim with a light hint, and I have had great trouble deriving this claim, any help would be appreciated -
Let ##R \in SO(N)## be an ##N##-dimensional rotation, then the...
I'm self-studying tensors from a book that doesn't have exercises. The book is Semi-Riemannian Geometry by Newman. To get a better feel for index manipulation, tensor results and calculations, I'm looking for a book that has many exercises in these topics.
I'd be grateful if those knowledgeable...
In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors, yet a (1,0) tensor is a vector and a (0,1) tensor is a dual vector. I must be missing something simple. Please explain.
Hello everyone! I'm currently self studying knot theory and I am at the point where I am looking at its relationship with other fields. I am a math and physics student, but my physics understanding is far behind my understanding of math. Hence, I would really like some help interpreting some...
This set of videos by eigenchris (separate playlists on Relativity and on Tensors) also looks interesting
and can help anyone interested in learning about these topics.
A while back I watched some of them and thought they could be helpful.
I like his presentation of one-forms.
(I've been...
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...
According to Cauchy's stress theorem, the stress vector ##\mathbf{T}^{(\mathbf{n})}## at any point P in a continuum medium associated with a plane with normal unit vector n can be expressed as a function of the stress vectors on the planes perpendicular to the coordinate axes, i.e., in terms of...
In ch. 13, pg.349 of Wald it's asked to prove that ##g_{AA'BB'} = \epsilon_{AB} \bar{\epsilon}_{A'B'}## is a Lorentz metric on ##V## (containing the real elements of the vector space ##Y## of type ##(1,0;1,0)## tensors). Given the basis ##t^{AA'} = \dfrac{1}{\sqrt{2}}(o^A \bar{o}^{A'} + \iota^A...
I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways:
##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
> **Exercise.** Let T1and T2be tensors of type (r1 s1)and (r2 s2) respectively on a vector space V. Show that T1⊗
T2can be viewed as an (r1+r2 s1+s2)tensor, so that the
> tensor product of two tensors is again a tensor, justifying the
> nomenclature...
What I’m reading：《An introduction to...
I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative
1) for scalar function;
$$\nabla_a\phi \equiv \partial_a\phi~~(1)$$
2) for vectors;
$$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$
$$\nabla_cV_a = \partial_cV_a -...
I am looking for math books that focus on geometrical interpretations. Sadly most of the modern books lack these interpretations and only consists out of theorems and proofs. It seems to me that most modern mathematicians are pure left-brain sequential thinkers that do not have a lot of...
Summary:: Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR?
Does the textbook Misner Thorne and Wheeler have all I need to understand tensors in order to learn GR? I have that textbook but never went through it. Tensors greatly intimidate...
Hello everyone. I was browsing through Amazon and found the aforementioned book by Theodore Frankel. As it is available at a relatively cheap price and covers a TON of material I was considering buying it for future use . Although the author says the prerequisites are only multivariable...
I have encountered two (?) definitions of the electric quadrupole moment. They are:
$$Q_{ij}=\frac{1}{2}\int \rho(\vec{x}')x'_i x'_j\,\mathrm{d}^3x'$$
and
$$Q_{ij}=\int (3x'_i x'_j-\delta_{ij}x'^2)\rho(\vec{x}')\,\mathrm{d}^3x'$$
I am trying to study radiation arising from the electric...
I try to solve but i have 1 step in the solution that I don't understand who to solve.
Below in the attach files you can see my solution, the step that I didn't make to prove Marked with a question mark.
thanks for your helps (:
Hi,
I'd to work with a model which represents a contact problem. I want to suppose that f_0=0 and f_2=0 where f_0 is a density of body forces and f_2 is a density of surface tractions .
I am mathematician so I don't know if the hypothesis that I'd to suppose will make the problem have a sense in...
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...
I read in the book Gravitation by Wheeler that "Any tensor can be completely symmetrized or antisymmetrized with an appropriate linear combination of itself and it's transpose (see page 83; also this is an exercise on page 86 Exercise 3.12).
And in Topology, Geometry and Physics by Michio...
There are a few different textbooks out there on differential geometry geared towards physics applications and also theoretical physics books which use a geometric approach. Yet they use different approaches sometimes. For example kip thrones book “modern classical physics” uses a tensor...
I am reading Group Theory in a Nutshell for Physicists by A. Zee.
I have big problems when learning chapter IV.1 Tensors and Representations of the Rotation Groups SO(N).
It reads
I can understand why ##D\left ( R \right )## is a representation of SO(3), but I hardly can see why the tensor T...
I am trying to understand the following:
$$
\epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp}
$$
Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices...
Can you contract any part of the stress energy tensor with the metric? Say if you had four components Tu1 and contracted that with g^u1 would that produce an invariant?
I know in 1939 Wigner published a theorem that all fields must be tensors from a couple of books, but can't find the proof anywhere. That obviously is an important result so does anyone know where I can find the proof? Another I haven't seen the proof of is the no interaction theorem. I wish...
I have read many GR books and many posts regarding the title of this post, but despite that, I still feel the need to clarify some things.
Based on my understanding, the contravariant component of a vector transforms as,
##A'^\mu = [L]^\mu~ _\nu A^\nu##
the covariant component of a vector...
Given a Weyl transformation of the metric ##g_{\mu\nu} \rightarrow g'_{\mu\nu} = e^{\Omega(x)} g_{\mu\nu}##, I'm trying to find the corresponding connection ##\Gamma'^{\lambda}_{\mu\nu}##, and from that ##-## via the Riemann tensor ##R'^{\lambda}_{\mu\nu\kappa}## ##-## the Ricci tensor...
Hello! I came across spherical tensors, and I am a bit confused about the way they are applied. For example, Pauli matrices, can be grouped together to form a rank 1 (vector) spherical tensor as ##(\sigma_-, \sigma_z, \sigma_+)##, which are the raising operator, the z projection operator and the...
This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is:
$$
T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...
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 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...