All Questions
Tagged with vector-analysis tensors
15
questions
7
votes
1
answer
3k
views
Divergence theorem for a second order tensor
I want to integrate by part the following integral in cylindrical coordinates
$$\int \vec{r} \times (\nabla \cdot \overline{T}) ~d^3\vec{r} $$
where $\overline{T}$ is a second order symmetric tensor ...
- 123
6
votes
2
answers
8k
views
Formula of the gradient of vector dot product
On Wikipedia in the article "Vector calculus identities"
(https://en.m.wikipedia.org/wiki/Vector_calculus_identities)
there are the following two formulas for computing the gradient of vector dot ...
- 61
5
votes
1
answer
319
views
What is the most general/abstract way to think about Tensors
In their most general and abstract definitions as Mathematical Objects :
A Scalar is an element of a field used to define Vector Spaces
A Vector is an element of a Vector Space.
Since a Scalar is ...
- 13.2k
4
votes
3
answers
8k
views
Product of Levi-Civita symbol is determinant?
I am confused with how can one write product of Levi-Civita symbol as determinant? I want to prove 'epsilon-delta' identity and found this questions answers it. But I am stuck at product of Levi-...
- 41
3
votes
0
answers
1k
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 ...
user87128
2
votes
1
answer
1k
views
$\nabla\times(\nabla\times \boldsymbol{A})$ using Levi-Civita
I want to prove that $\nabla\times(\nabla\times \boldsymbol{A}) = \nabla(\nabla\cdot\boldsymbol{A}) - \nabla^2\boldsymbol{A}$ using the Levi Civita. This is solved considering the $i$-th component of $...
- 4,655
1
vote
2
answers
1k
views
Levi civita and kronecker delta properties?
I'm trying to grasp Levi-civita and Kronecker del notation to use when evaluating geophysical tensors, but I came across a few problems in the book I'm reading that have me stumped.
1) $\delta_{i\,j}...
- 11
9
votes
2
answers
2k
views
Special case of the Hodge decomposition theorem
I am trying to prove the following special case of the Hodge decomposition theorem in differential geometry for an $n$ component vector field $V_i$ in $\mathbb{R}^n$. I have very little knowledge of ...
user23238
6
votes
1
answer
8k
views
How can you calculate distance in plane polar coordinates using the metric tensor
I'm trying to develop some intuition about the metric tensor and how it can be used to calculate distances/angles in curvlinear space by using the very simple example of a 2D polar surface.
The ...
- 63
4
votes
2
answers
413
views
Proving $A\times(\nabla\times B)+B\times(\nabla\times A)=\nabla(A\cdot B)-(A\cdot\nabla)B -(B\cdot\nabla)A$ with Einstein summation
So, I'm seeking to prove the below identity, for $A,B$ vectors fields in $\mathbb{R}^3$:
$$A \times (\nabla \times B) + B \times (\nabla \times A) = \nabla (A \cdot B) - (A \cdot \nabla)B - (B \cdot \...
- 45.9k
2
votes
1
answer
149
views
Extremely complex vector-matrix expression and its differentiation by vector
Given:
$Q=R_z(\psi)R_y(\xi)R_x(\phi)$ - rotation matrix
$\boldsymbol{\theta}=\left[\begin{array}{@{}c@{}} \phi \\ \xi \\ \psi
\end{array} \right]$ - vector of angles
$p=Q\left[\begin{array}{@{}...
- 731
2
votes
2
answers
9k
views
Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$
Question: Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$ where $\epsilon_{ijk}$ is the Levi-civita symbol and $\delta_{ij}$ is the Kronecker delta symbol.
My attempt: $\epsilon_{ijk}$ assumes ...
- 24.5k
0
votes
2
answers
3k
views
Introducing new indices with tensor/index notation
I understand where the $k$ comes from in $\varepsilon_{klm}$, however why do we need to introduce $l,m$ rather than continuing with $i,j$,
i.e $\varepsilon_{kij}b_ic_j$
- 1,257
0
votes
1
answer
762
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.
...
- 3,036
0
votes
1
answer
161
views
Relationship between the set of elementary matrices and 2-tensors
I apologize for the wordiness of this question, but it's largely conceptual (and might not even make sense).
This is a question about Analysis 2 (particularly, analysis on manifolds). I've been ...
- 35