1
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0answers
46 views

What exactly do the terms of $(f \circ g)'''$ mean?

Say, $g: X\to Y$ and $f: Y\to Z$ are smooth. One can find $(f \circ g)'''$ by using the FaĆ  di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
1
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0answers
18 views

Hessian matrix of $g\circ f$

Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$ $$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial ...
3
votes
2answers
166 views

Green's first identity

Good morning/evening to everybody. I'm interested in proving this proposition from the Green's first identity, which reads that, for any sufficiently differentiable vector field $\mathbf{\Gamma}$ and ...
0
votes
0answers
18 views

Tensor notation and the “Zero-Value Theorem”

In the following picture: taken from Martin Sadd's book on elasticity, I am having trouble understanding the "zero-value theorem". I can't understand why this theorem is true. For example, when ...
1
vote
1answer
55 views

Why is this true: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec v)+\vec v \cdot (\nabla \cdot S)$?

I am doing fluid mechanics and I don't understand a particular step that is being used. It is the following step which I don't understand: $\nabla \cdot (S\cdot \vec v )=S:(\nabla \otimes \vec ...
2
votes
1answer
105 views

Using the Levi-Civita alternating tensor and suffix notation to concisely write the vector product rule.

I am reading through a section on vector calculus in an electromagnetism book and it has started to use suffix notation and the Levi-Civita alternating tensor in order to prove some identities. Some ...
1
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1answer
102 views

Why is this true:$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $

Can someone help me why the following is true: $$ \nabla \cdot (\vec V \otimes \vec V)=(\vec V\cdot \nabla ) \vec V +\vec V(\nabla\cdot \vec V) \;\;? $$ I've thought of the following relation to be ...
0
votes
2answers
92 views

Intuition behind $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$ (Levi Civita)

Let $\vec{e_i}$ denote a unit vector. Then we can write: $\vec{e_i} \times \vec{e_j}=\epsilon_{ijk} \vec{e_k}$, where $\epsilon_{ijk}$ is the Levi Civita symbol. Can someone intuitively explain me ...
1
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1answer
630 views

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 ...
3
votes
1answer
41 views

$\nabla \varphi \overset{?}{=} \nabla \cdot \varphi \bar{\bar{I}}$ where $\varphi$ is scalar, $\bar{\bar{I}}$ is identity tensor

I am trying to determine if these two are equivalent. I have a function written with both terms, and this is the only discrepancy. The gradient increases the rank of the scalar to a vector, while the ...
4
votes
1answer
243 views

Derivation or Intuition of Formula for Levi-Civita Symbol

http://www.ees.nmt.edu/outside/courses/GEOP523/Docs/index-notation.pdf spouted off and threw out with no motivation $$\epsilon_{ijk} = \frac{1}{2}(i - j)(j - k)(k - i) \, \forall \, \, k \in \{1, 2, ...
1
vote
1answer
53 views

The gradient of a function is an alternating one-tensor

I'm currently reading Spivak's Calculus on Manifolds and I seem to have hit a snag in Chapter Four: Integration on Chains. Spivak develops tensors, vector fields, alternating tensors and differential ...
3
votes
1answer
75 views

Simplifing formulas using tensor notation

Im trying to symplify formulas like: $$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$ or something more strange like: ...
1
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1answer
114 views

Is the third order term of a Taylor approximation a covariant tensor of rank 3 restricted to (h,h,h)?

Recently reading Peter Lax's linear algebra text, he had a very concise way of writing the Taylor approximation up to second order: $f(x+h) = f(x) + l(h) + \frac{1}{2}q(h) + \|h\|^2 \epsilon(\|h\|)$, ...
2
votes
3answers
289 views

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 ...
0
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1answer
78 views

Help needed with tensors [duplicate]

Possible Duplicate: An Introduction to Tensors Recently I came across the concept of tensors and heard it is very difficult to understand. Is there a ...
0
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1answer
2k views

What is the divergence of a matrix valued function?

According to Wikipedia: The divergence of a continuously differentiable tensor field $\underline{\underline{\epsilon}}$ is: ...
0
votes
1answer
213 views

gradient of row vector multiplied by scalar

I'm trying to re-write $v (u x)$ where $v$ and $u$ are row vectors and $x$ is a column vector as some expression $M x$ (or $\bar{v}x$, etc.). The motivation is because I'm trying to compute the ...
1
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1answer
227 views

Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...