3
votes
2answers
35 views

Properties and notation of third-order (and higher) partial-derivatives

This question has been bothering me for quite a while and I still haven't found a satisfying answer anywhere on the internet or in any of my books (which may not be that advanced, mind you...). Since ...
0
votes
1answer
38 views

Question about index notation on partial derivatives.

I've been studying quantum field theory a little bit and I've encountered a notation like the following: $$\mathcal{D}_{x,x'}=\frac{\partial}{\partial x^\mu}\frac{\partial}{\partial ...
0
votes
2answers
60 views

Multiplication and derivation of 3D matrix

I have $A(q)=\begin{bmatrix}q_1 &q_2 & q_3\\ 2q_1 &3q_2 & 4q_3\\ 2q_1 &3q_1 & 10\\ \end{bmatrix}\tag 1$ $ q= {\left(\begin{array}{c}q_1\\q_2\\q_3\\q_4\\q_5\\q_6 ...
1
vote
0answers
22 views

Characterization of the derivative as a tensor field

I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional ...
1
vote
1answer
60 views

Integral/Vector calculus $\oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S.$

I am trying to show that $$ \oint_{\partial S} u \vec \nabla v \cdot d \vec \lambda=\int_S (\vec \nabla u)\times (\vec \nabla v)\cdot d\vec S $$ using Levi Cevita notation methods only. The Levi ...
0
votes
0answers
27 views

Misleading tensor notation for Jacobian inverse?

In Schutz, Geometrical Methods of Mathematical Physics, is written a Jacobian coordinate transform $\Lambda$, $$ \Lambda^i_j = \frac{\partial x^i}{\partial y^j} $$ The inverse matrix is written $$ ...
0
votes
1answer
29 views

jacobian times its inverse - should be identity

Here's an easy one. A Jacobian is $\frac{dx^i}{dy^j}$. The inverse is $\frac{dy^j}{dx^k}$. So, in tensor notation, $\frac{dx^i}{dy^j} \frac{dy^j}{dx^k} = \frac{dx^i}{dx^k} = \delta^i_k$ Now ...
1
vote
0answers
51 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 Faà di Bruno's formula: $$(f \circ g)''' =(f'''\circ g)(g')^3 + 3(f''\circ g)g'g'' + (f'\circ g)g'''$$ But my ...
2
votes
0answers
27 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
224 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
21 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
69 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
2answers
166 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
vote
1answer
129 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
103 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 ...
2
votes
1answer
1k 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
43 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
269 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
62 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
80 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
vote
1answer
133 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
356 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
votes
1answer
81 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
votes
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
224 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
vote
1answer
230 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 ...