Tagged Questions
1
vote
3answers
27 views
notation question (bilinear form)
So I have to proof the following:
for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form.
...
1
vote
0answers
23 views
Solving tensor Identities
For my homework I am required to prove the following identity. Here $\bf{u}$ is a vector.
$\nabla \bf{uu} = u(\nabla.u) + (u.\nabla)u$
Only thing I understand in this equality is lest hand side is ...
1
vote
1answer
58 views
How to solve a tensor differential equation?
Essentially, How does one solve the tensorial differential equation $$\frac{dx^a}{d\tau}=A^a{}_bx^b$$
where $x^a$ is a 4-vector and $A^a{}_b$ is a $(1,1)$ tensor.
The original Problem
How does ...
0
votes
0answers
53 views
What is mathematics (in physics) of this tensor equation?
I want to know what is $C^{a}C_{a}$ if $C^{a}=A^{a}+B^{a}$ and $C_{a}=A_{a}+B_{a}$?
a. this one $A^{a}A_{a}+B^{a}B_{a}$ or
b. this one $(A^{a}+B^{a})(A_{a}B_{a})$
2
votes
1answer
37 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
59 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{ ...
0
votes
0answers
80 views
Derivative of a vector with respect to a matrix
I am at an impasse. I don't know if homework is allowed on here or not, so if it isn't, someone delete this.
Given:
$H_{\gamma} = C_{\beta \beta} v_{\gamma} + C_{\beta \varepsilon} C_{\varepsilon ...
1
vote
1answer
84 views
Matrix representing $\Lambda^k$(A)
Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map.
Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the ...
1
vote
2answers
619 views
Prove the determinant of a tensor is invariant
Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$.
Prove that the determinant of the tensor $T$
$\det T=\frac{Tu.(Tv ...
