All Questions
16
questions
2
votes
1
answer
75
views
geometric interpretation on covariant derivateve in curvilinear coordinates.
I'm having trouble understanding what does the covariant derivative do in a coordinate system where we have changing basis vectors. I always thought it was giving us the change in coordinates while ...
- 379
1
vote
1
answer
140
views
Covariant and partial derivative of a vector field (not component)
Is the covariant derivative of a vector field (not the components of a vector) same as the partial derivative?
I am adding a screenshot from page 69 from General Relativity: An introduction for ...
- 109
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
1
answer
98
views
Expression for Rank 2 Tensor in Vector Notation
How does one write the following expression
$D_{jk} (r_k \delta_{ij} - r_{i}\delta_{jk} - r_j \delta_{ik})$
in matrix notation? Is this just
$\textbf{D} (\textbf{r} \times \textbf{I})$?
- 3,005
0
votes
0
answers
223
views
Gradient of a vector field in curvilinear coordinates
What is the formula for calculating the gradient of a vector field if the field is expressed in terms of curvilinear coordinates like spherical or cylindrical systems?
0
votes
0
answers
23
views
Identity relating $V \circ \nabla F$ and $V( \nabla \circ F)$?
I’ve been trying to derive the Leibniz Integral Rule 1 in 3D and I’ve nearly got it, but where Wikipedia shows $V \circ \nabla F$, my approach gives $V( \nabla \circ F)$. I’m thinking that the two ...
user564215
0
votes
0
answers
41
views
Two questions about vector equations
Q1. How to write the following equation in a form of $\vec{a}= something$?
From the question posted in Physics SE, I found the following process is wrong.
For any $ \vec{v}\ne\vec{0}$
$$ \vec{v}\...
- 45
-1
votes
1
answer
45
views
Prove of some vector's differential relations [closed]
I want to know how to prove these two relations:
$(V⋅∇)V=\frac12∇(V⋅V)−V×(∇×V)$
$(∇.sv)=(∇s.v)+s(∇.v)$
[These relations are from Bird's Transport Phenomena]
0
votes
1
answer
117
views
Tensor and Vector Notation
I'm given the tensor $X^{\mu\nu}$ and vector $V^\mu$ of the form
$$X^{\mu\nu} = \begin{bmatrix}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
...
- 603
3
votes
2
answers
493
views
Need help proving $\vec{u}\times \vec{\omega}=\vec{\nabla}(\frac{1}{2}\vec{u}\cdot \vec{u})-\vec{u}\cdot \vec{\nabla}\vec{u}.$
I've set out to prove $\vec{u}\times \vec{\omega}=\vec{\nabla}(\frac{1}{2}\vec{u}\cdot \vec{u})-\vec{u}\cdot \vec{\nabla}\vec{u}.$
This is my try:
$$\vec{u}\times \vec{\omega}=\frac{1}{2}\vec\nabla\...
- 1,675
1
vote
1
answer
332
views
Vector proof using Levi-Civita notation
Can anyone expand/simplify this proof: I am unsure why it is the partial derivative with respect to $x_i$, is there no $x_j$ or $x_k$?
$$
\begin{align}
\underline{\nabla} \cdot (\underline{A} \times \...
- 1,111
0
votes
1
answer
101
views
Einstein notation difficulties
I'm just learning the Einstein index notation, and came across this derivation in a textbook. I couldn't follow the steps. Can someone please help me out?
The first order differential equation:
$$\...
1
vote
1
answer
2k
views
Finding the Gradient of a Vector Function by its Components
In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field.
$$...
- 13.2k
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
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