1
vote
1answer
34 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...
0
votes
1answer
57 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
0
votes
1answer
31 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
0
votes
0answers
45 views

Relation between componenet and algebraic definition of covariant vectors

I studied contravariance and covariance concepts in following way: For any vector if we get its components by parallelogram way we achieve contravariant components, and if we want to get its ...
0
votes
0answers
65 views

Gradient is covariant or contravariant?

I read somewhere people write gradient in covariant form because of their proposes. I think gradient expanded in covariant basis i , j , k so by invariance nature of vectors, component of gradient ...
0
votes
1answer
47 views

A question about vector fields and divergence

I am reading the paper http://www.goshen.edu/physix/mathphys/gco/TensorGuideAJP.pdf in order to gain a basic understanding about tensors. I had some difficulties about understanding some definitions. ...
2
votes
2answers
48 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
0
votes
2answers
52 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
3
votes
1answer
94 views

Why the space of skew-symmetric tensors $\Lambda^{n}V$ is a one dimensional if $dim(V)=n$

While reading Liviu Nicolaescu Lectures on the geometry of manifolds, I came accross the notion of "determinant line": Definition: Lev $V$ be an n-dimensional R-vector space. The one dimensional ...
2
votes
2answers
88 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
6
votes
1answer
334 views

Vectors, Basis, Dual Vectors, Dual Basis and Tensors

I'm trying to understand tensors and I know they have something to do with the basis and the dual basis of a vector space and a dual space. First I will give a concrete example to make clear what I ...
9
votes
2answers
792 views

Do I understand metric tensor correctly?

So I've been studying vectors and tensors, and I'm trying to understand metric tensors. As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors ...
3
votes
1answer
465 views

Basis of vector fields on manifold

For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of ...
4
votes
1answer
49 views

Is a $1^{\mathrm{st}}$ rank tensor identical to a vector?

As far as I know, we define vectors as elements of a vector space, then there is an isomorphism (by choosing a basis) from the vector space to tuples of components in some field, $\mathbb{F}$ say. ...
1
vote
0answers
70 views

To show that something is a four-vector

I hope this question is not too inane... it would be really helpful for me to have this cleared up. I want to know what I need to show to demonstrate that something is a four-vector. I have checked ...
0
votes
1answer
299 views

Dot product between two vectors or vector and 1-form?

When we take a dot product between two vectors of a vector space, we actually "act" by a 1-form (dual vector) on a vector. So why most books define the dot product between vectors? Of course with the ...
3
votes
3answers
71 views

Definition of a tensor field

Could anybody explain to me the following: If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$ where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative, then $T_{ij}$ is a tensor field. ...
1
vote
2answers
101 views

Action of $\mathfrak{sl}({V})$ in tensor spaces

what is the natural action of $\mathfrak{sl}({V})$ in tensor spaces ?
2
votes
2answers
37 views

$4$-vectors and indices

I am reading through some material about $4$-vectors. And came across the following for which an explanation woud be greatly appreciated. The index for $\partial_\alpha$ can be raised giving ...
1
vote
1answer
116 views

basic vector being hermitian

If the space has a mixed metric signature, not all the basis vectors are Hermitian. Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose conjugate is, ...
0
votes
1answer
80 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...