All Questions
8
questions
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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:
$$div(\vec{u})=\vec{a}^i\cdot\frac{\partial \vec{a}_ku^k}{\partial x^i}=\frac{\partial u^i}{\...
- 379
4
votes
2
answers
413
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Proving $A\times(\nabla\times B)+B\times(\nabla\times A)=\nabla(A\cdot B)-(A\cdot\nabla)B -(B\cdot\nabla)A$ with Einstein summation
So, I'm seeking to prove the below identity, for $A,B$ vectors fields in $\mathbb{R}^3$:
$$A \times (\nabla \times B) + B \times (\nabla \times A) = \nabla (A \cdot B) - (A \cdot \nabla)B - (B \cdot \...
- 45.9k
0
votes
1
answer
241
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Divergence of a Tensor Field
Given a tensor field $\hat{\tau}$, I wish to calculate $\nabla\cdot\hat{\tau}$.
My first question: is this actually the divergence of the tensor field? Wikipedia seems to differentiate between div$(\...
- 1,019
3
votes
1
answer
6k
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Calculating the Divergence of a Tensor
I am working through a fluid dynamics paper and came across this equation:
$$ \frac{\partial \vec{v}}{\partial t} + \vec{v}\cdot\nabla\vec{v}=\nabla\cdot T - \frac{1}{\rho}\nabla \phi\tag1$$
where T ...
- 1,019
3
votes
1
answer
148
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Proving $(\nabla \times \mathbf{v}) \cdot \mathbf{c} = \nabla \cdot (\mathbf{v} \times \mathbf{c})$ using cylindrical coordinates
Assuming the form of divergence in polar coordinates is known, I am attempting to use the following definition of the curl of a vector field to determine the form of the curl in cylindrical ...
- 133
0
votes
1
answer
51
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How do I reexpress the equation $\nabla \times (\nabla \times gs)) \times (\nabla \times \nabla(f\nabla \cdot t))$?
How do I go about reexpressing
$\nabla \times (\nabla \times bs)) \times (\nabla \times \nabla(c\nabla \cdot t))$
where s and t are vector properties and b and c are scalar.
I don't know where to even ...
0
votes
1
answer
167
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Gradient in tensor form
I found a problem which had $$\partial_i (A_i \vec{G})= (\vec{\nabla} .\vec{ A} )\vec{G}+ (\vec{A}.\nabla) \vec{G} $$ but my problem is what does $$\partial_i (A_i \vec{B})$$ even mean? it doesn't ...
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1
vote
2
answers
670
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divergence of gradient of scalar function in tensor form
I found simple expression in tensor notation for a divergence of product vector and gradient of scalar function:
$$\operatorname{div}(\mathbf{j}) = 0 \text{, where } \mathbf{j} = \mathbf{m}\times \...
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