Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...
1
vote
1answer
15 views
Prove that a tensor field is of type (1,2)
Let $J\in\operatorname{end}(TM)=\Gamma(TM\otimes T^*M)$ with $J^2=-\operatorname{id}$ and for $X,Y\in TM$, let
$$N(X,Y):=[JX,JY]-J\big([JX,Y]-[X,JY]\big)-[X,Y].$$
Prove that $N$ is a tensor field of ...
2
votes
1answer
17 views
Contraction of the second Bianchi identity
The second Bianchi identity is
$${R^a}_{b[cd;e]}=0$$
And contracting it with respect to $a$ and $e$ we get
$${R^a}_{b[cd;a]}=0 \Leftrightarrow $$
$${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$
What I don't ...
0
votes
0answers
18 views
Norm of tensor object
Suppose I have a $3\times2 \times 2$ tensor object $M$. What is then $|M|$ ?
Thank you for your support!
1
vote
1answer
53 views
Quotient theorem for tensors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
2
votes
1answer
46 views
Parallel Transport along a curve
We had this homework assignment for our geometry course, and we couldn't figure it out, any ideas on how to do this:
Consider the Poincare model of Lobachevsky plane,
$H^2=\left\lbrace{ ...
1
vote
0answers
40 views
Riemannian curvature and its application on covariant derivative of tensors
This identity can be generalized to get the commutators for two
covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
2
votes
1answer
22 views
Regarding the definition of covariant derivative and its use on basis vector fields
we find that for general vector fields ${\mathbf v}= v^ie_i$ and ${\mathbf u}= u^je_j$ we get :$\nabla_{\mathbf v} {\mathbf u} = \nabla_{v^i {\mathbf e}_i} u^j {\mathbf e}_j = v^i \nabla_{{\mathbf ...
0
votes
0answers
13 views
what is the status of the theory of multilinear systems of equations?
What is the current status of the theory of multilinear systems of equations?
I have a particular interest for multilinear homogeneous systems of the form
$A_1 \otimes \cdots \otimes A_r) (x_1 ...
1
vote
1answer
50 views
Gradient with respect to a matrix variable
I want to find the gradient of the function $\mathcal{F}_1$ with respect to the matrix $\mathbf{X}$ (differentiate with respect to $\mathbf{X}$):
$$
\mathcal{F}_1 (\mathbf{X}; \mathbf{\lambda})= ...
0
votes
1answer
15 views
Regarding confusion of basis tensors and the usage of tensors.
Let us for example give a tensor example of following: $X = X^i \partial_i$. According to mny knowledge, in this case $\partial_i$, basis, is treated as tensor (otherwise, $X$ as tensor won't be ...
1
vote
1answer
25 views
Finding an “inverse” of a deviatoric tangent
I have have a material model, defining the deviatoric stress for a nonlinear fluid:
$\boldsymbol{\sigma}_{\mathrm{dev}} = f(\dot{\boldsymbol{\varepsilon}}_{\mathrm{dev}})$
Now I wish to find the ...
0
votes
0answers
31 views
An exact sequence from tensors
Let $V$ be a vector spaces. Why is the following sequence exact?
$S_{[table]}V \rightarrow V\otimes \Lambda^{2}V \rightarrow \Lambda^{3}V$
where suffix table is a diagram of a table with first row [ ...
0
votes
1answer
57 views
Proving the symmetry of the Ricci tensor?
Consider the Ricci tensor :
$R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho}
-\partial_{\nu}\Gamma_{\rho\mu}^{\rho}
+\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda}
...
-1
votes
0answers
84 views
Some tensor questions
I am trying to self read tensor geometry. I am stuck at the following questions that will help me build greater intuition and insight. Could someone guide me?
Let $V$ be a linear space.
Why is the ...
0
votes
0answers
32 views
Einstein notation non-repeating indices
I forget the rule for Einstein notation. If I have something like the gradient:
$$\vec\nabla f = \frac{\partial f}{\partial x_i} = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial ...
