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.
In Wald's "General Relativity", in his section on abstract tensor notation, he let's g_{ab} denote the metric tensor. When applied to a vector v^a, we get a dual vector, because g_{ab}(v^a, \cdot) is just a dual vector. Okay cool. But then he says that this dual vector is actually g_{ab}v^b...
They are being 2 by 2 matrices and I being the identity. Physically they are Pauli matrices.
1. Is $$((A\otimes I\otimes I) + (I\otimes A\otimes I) + (I\otimes I\otimes A))\otimes B$$
= $$(A\otimes I\otimes I)\otimes B + (I\otimes A\otimes I)\otimes B + (I\otimes I\otimes A)\otimes B$$? I...
What is meant by taking the tensor product of vectors? Taking the tensor product of two tensors is straightforward, but I am currently reading a book where the author is talking about tensor product on tensors then in the next paragraph declares that tensors can then be constructed by taking...
Hi all!
I'm really interested in the physical interpretation of the Riemann and metric tensors. Is there any program that let's you type in a Riemann or metric tensor (or even better, an Einstein tensor) and then gives you an image of how the space would look (i.e. the curved grid lines)?
Thanks!
Homework Statement
Verify that ##\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}+\partial_{\lambda}F_{\mu\nu}## is equivalent to ##\partial_{[\mu}F_{\nu\lambda]}=0##,
and that they are both equivalent to ##\tilde{\epsilon}^{ijk}\partial_{j}E_{k}+\partial_{0}B^{i}=0## and...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ...
The relevant text in...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get an understanding of an aspect of Example 10.11 and Definition 10.7 in Section 10.3 ...
The relevant text in...
I am reading Dummit and Foote: Abstract Algebra (Third Edition) ... and am focused on Section 11.5 Tensor Algebras. Symmetric and Exterior Algebras ...
In particular I am trying to understand Theorem 31 but at present I am very unsure about how to interpret the theorem and need some help in...
I am reading Dummit and Foote: Abstract Algebra (Third Edition) ... and am focused on Section 11.5 Tensor Algebras. Symmetric and Exterior Algebras ...
In particular I am trying to understand Theorem 31 but at present I am very unsure about how to interpret the theorem and need some help in...
Homework Statement
Maxwell's Lagrangian for the electromagnetic field is ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}## where ##F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}## and ##A_{\mu}## is the ##4##-vector potential. Show that ##\mathcal{L}## is invariant under gauge...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...
If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Example 10.1 in Section 10.3 ...Example 10.1 plus some preliminary definitions reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Definition 10.5 in Section 10.3 ...Definition 10.5 plus some preliminary definitions reads as...
When finding the Von Mises of given a stress tensor who's only element is a single shear component (τ):
\begin{bmatrix}
0 & τ & 0\\
τ & 0 & 0\\
0 & 0 & 0
\end{bmatrix}
the result is simply √3×τ. Is the Von Mises criterion not valid when considering a single component as in this example? I...
Homework Statement
The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by
$$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$
Show...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.3 The Tensor Algebra ... ...
I need help in order to get a basic understanding of Theorem 10.8 which is a Theorem concerning the direct sum of a family of subspaces as a...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ...
My apologies if my...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help in order to get a basic understanding of Theorem 10.2 regarding the basis of a tensor product ... ...
My apologies if my...
A particle in a 1-D Hilbert space would have position basis states ## |x \rangle ## where ## \langle x' | x \rangle = \delta(x'-x) ## A 3-D Hilbert space for one particle might have a basis ## | x,y,z \rangle ## where ##\langle x', y', z' | x,y,z \rangle = \delta(x'-x) \delta (y-y') \delta(z-z')...
Hi,
I understand stress, strain but when it moves on to 3 dimension anisotropic materials using tensors and stiffness matrices I get confused with einstein's notation. can someone please help me out in this regard to undrstand how stiffness and compliance matrices get reduced for monoclinic...
In the middle of the below paragraph: "only if the shift vector ##R## is along one of the principal axes relative to the center of mass will the difference tensor be diagonal in that system." I suppose the difference tensor means new inertial tensor ##-## old inertial tensor.
That means the new...
Hello,
Is the covariant conservation of the matter energy momentum tensor Tμν ; μ = 0 also valid in a theory of gravity having an action for the gravitational field different from the Einstein Hilbert action ?
I'm asking because in GR the einstein field equations require Tμν= Gμν
where Gμν;μ=0...
I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ...
I have a basic question regarding the extension of the scalars ...
Dummit and Foote's exposition regarding extension of the scalars reads as...
I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ...
I have a basic question regarding the extension of the scalars ...
Dummit and Foote's (D&Fs) exposition regarding extension of the scalars reads as...
Is there a simple geometric interpretation of the Einstein tensor ? I know about its algebraic definitions ( i.e. via contraction of Riemann's double dual, as a combination of Ricci tensor and Ricci scalar etc etc ), but I am finding it hard to actually understand it geometrically...
Homework Statement
Suppose A and B are vectors. Show that the object Q with nine components Qij=AiBj is a tensor of rank 2.
Homework Equations
A tensor transforms under rotations (R) as a vector:
Tij'=RinRjmTnm
The Attempt at a Solution
I wanted to just create the matrix, but I don't know how...
I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector.
I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$
I can't figure out a way to get the required...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the proof...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 ... ... basically I do not know what is going on in the second part of the...
In the transformation of tensor components when changing the co-ordinate system, can someone explain the following:
Firstly, what is the point in re-writing the indicial form (on the left) as aikTklajl? Since we're representing the components in a matrix, and the transformation matrix is also...
I'm having trouble with the mathematics of tensor products as applied to Bell states.
Say I have the state
\begin{align*}
\left|\psi\right> &= \frac{1}{\sqrt{2}} \left(\left|0\right>_A \otimes \left|0\right>_B + \left|1\right>_A \otimes \left|1\right>_B\right)
\end{align*}
How would the...
I read Weyl tensor helps on propagating gravitational effects. Ricci is local depending on mass energy at that point and would vanish at other points. Weyl propogates the gravity effects (for example gravity at any point between Earth Moon is due to Weyl Tensor). I didn't quite get it...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ...The relevant part of Theorem 10.3...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.2 Properties of Tensor Products ... ...
I need help with an aspect of the proof of Theorem 10.3 regarding a property of tensor products ... ... The relevant part of Theorem...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ...Theorem 10.2 reads as follows:
I do not...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with an aspect of Theorem 10.2 regarding the basis of a tensor product ... ... Theorem 10.2 reads as follows:I do not...
Homework Statement
This is Exercise 15.2 in MTW - See attachment
Homework Equations
See attachment
The Attempt at a Solution
[/B]
My attempt at a solution is also in the attachment.
Are my initial assumptions OK? If not can someone nudge me in the right direction.
If my initial...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...The relevant part of...
I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ... The relevant part of...
Homework Statement
As the title says, I need to show this. A conformal transformation is made by changing the metric:
##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}##
Homework Equations
The Weyl tensor is given in four dimensions as:
##...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with a further aspect of the proof of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure...
I am reading Anthony W. Knapp's book: Basic Algebra in order to understand tensor products ... ...
I need some help with an aspect of Theorem 6.10 in Section 6 of Chapter VI: Multilinear Algebra ...
The text of Theorem 6.10 reads as follows:
The above proof mentions Figure 6.1 which is...