All Questions
Tagged with vector-analysis tensors
144
questions
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34
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Changing coordinates of $2$nd order partial operators
Let's work in $\mathbf R^n$. If we want to change coordinates $\mathbf x\to\mathbf r$, with them related like
$$
\mathbf x=\mathbf x(\mathbf r)
$$
Then, the second order generic partial operator in ...
- 2,638
0
votes
0
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115
views
tensor identity
The tensor $t$ is defined in terms of a scalar $\varrho$ and the two vectors $v$ and $u$ (and derivatives) as follows:
$$t_{i j}:=\varrho v_i u_j-\frac{1}{2} \varrho \varepsilon_{i lm} v_l (\...
3
votes
2
answers
404
views
Help with the gradient in different co-ordinate systems
Let $L(x,y)$ be the linear Taylor series expansion of some function $f(x,y)$. This can be written as $$L(x,y)=f(x_0,y_0)+f_x(x-x_0)+f_y(y-y_0)$$ Or in more compact form as $$L(x,y)=f(x_0,y_0)+\nabla f ...
- 111
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1
answer
57
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$\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$ using Levi-Civita
I need to prove the following equality: $\nabla\cdot(f\vec{g})=f\nabla\vec{g}+\vec{g}\cdot\nabla f$.
I know that proving this equality using the properties of $\nabla$ and $(\cdot)$ is easy, what the ...
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 ...
- 15.5k
25
votes
7
answers
11k
views
How would you explain a tensor to a computer scientist?
How would you explain a tensor to a computer scientist? My friend, who studies computer science, recently asked me what a tensor was. I study physics, and I tried my best to explain what a tensor is, ...
- 863
2
votes
1
answer
67
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Differential surface element and nabla operator
If we have the vector field $\vec{u}=\vec{A}\times \vec{v}$, where $\vec{A}=\text{const.}$ and we integrate over some closed curve, by using Stokes' theorem we get:
$$
\begin{align}
\oint_{\partial S}...
- 2,743
0
votes
1
answer
44
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If $S$ is a symmetric matrix, then rewriting $\int{S:\nabla \phi} dx$
I am trying to prove that:
If $S$ is a symmetric matrix, then one can rewrite $\int{S:\nabla \phi} \text{dx}$ as $\int{S:D(\phi)} \text{dx}$, where $D(\phi)$ is the symmetric gradient of $\phi$.
Any ...
- 35.1k
1
vote
1
answer
104
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Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz)
I can't figure it out about the Exercise 5.22 in the book Geometrical Methods of Mathematical Physics (by Bernard Schutz).
Could any one give me a help? Thanks.
($\bar{V}$ means vector V. )
Exercise ...
- 1,652
0
votes
2
answers
119
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What is the gradient in this skewed coordinate system?
Consider two bases $e_i$ and $f_i$ of $\mathbb{R}^2$ defined by:
$$\begin{aligned}
(f_1,f_2) &= (e_1,e_2)\cdot F\\
\begin{pmatrix}1&-1\\1&1\end{pmatrix} &= \begin{pmatrix}1&...
- 1,652
1
vote
1
answer
136
views
Analyzing the Hessian of this function
The problem I'm looking at is, given a matrix $A$ of size $m\times n$ where $A_{ij}\ge 0$ minimize $f(x, y) = \|A - xy\|^{2}_{F}$ where $x$ is a column vector of length $m$, $y$ is a row vector of ...
- 479
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votes
0
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52
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Intuition about the divergence of a vector field in non-orthogonal basis
My textbook defines the divergence of a vector field in a non orthogonal constant basis the following way:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
1
vote
0
answers
45
views
How to integrate by parts the vectorial product between a vector and a gradient [closed]
I am having a problem trying to check if the identity below should be positive or negative.
$$ \int\; \boldsymbol{A} \times (\boldsymbol{\nabla} a ) = \pm \int (\boldsymbol{\nabla} \times \boldsymbol{...
- 118k
2
votes
2
answers
93
views
Derivation or intuition on the covariant derivative for higher rank tensors
So the derivation in my textbook for the covariant derivative of a vector field $\vec{u}$ in curvilinear coordinates $\xi^k$ is the following:
$$\frac{\partial \vec{u}}{\partial\xi^j}=\frac{\partial (...
- 6,833
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0
answers
51
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Differential operators of tensor fields. Hamilton operator
The very first thing my textbook says is that the Hamilton operator is defined as:
$$\vec{\nabla}=\vec{a}^i\nabla_i$$
Where $\nabla_i$ is the covariant derivative and " $\vec{a}^i$ is the ...
- 379