All Questions
Tagged with vector-analysis tensors
144
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3
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What is the mathematical nature of a rotation matrix?
I have a naive question: what is the mathematical nature of a rotation matrix?
Is a rotation matrix a tensor ?
EDIT: if a rotation matrix is fundamentally a tensor, what is its (n, m) notation?
- 1,445
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1
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Prove: ∇⋅ϕF = ϕ∇⋅F + F⋅∇ϕ
I am asked to prove this identity using tensor notation. However, I am not sure where to even begin the problem.
- 1,557
4
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1
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Divergence of stress tensor in momentum transfer equation
Let suppose that we work in a 2D cartesian coordinates. what will be x and y components of
$\nabla.\left[-p I+\mu \left(\nabla \text{u}+(\nabla \text{u})^T\right)-\frac{2}{3} \mu (\nabla.\text{u})
I \...
- 4,358
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2
answers
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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
2
votes
4
answers
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Vector Calculus - Curl of Vector
I'm asked to prove the following identity, using index notation:
$(\nabla\times A)\times A=A \cdot\nabla A - \nabla(A \cdot A)$
However, when I work it out, I find that the actual solution should ...
- 21
1
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2
answers
969
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How do I calculate numerically a tensor in polar coordinates?
You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid?
a) fit a (spline) surface, ...
- 111
1
vote
1
answer
183
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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, therefore, ...
- 905
7
votes
2
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vector/tensor covariance and contravariance notation
As I looked over the Wikipedia article on covariance and contravariance of vectors and $\mathbf{v}=v^i\mathbf{e}_i$ is said as a contravariant vector while $\mathbf{v}=v_i\mathbf{e}^i$ is said as ...
- 905
1
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1
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Index/Einstein notation to derive Gibbs/Tensor relations
In a few continuum classes I have seen indicial notation used to derive relations in Gibbs notation. However, Gibbs notation is valid for all coordinates while indicial notation is valid only for ...
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