Cryo
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Justifying the "Physicist's method" for ODEs using differential forms
7 votes

I think I may be the 'physicist' in question, but I'll give it a go. $dy$ is a one-form on $\mathbb{R}^2$, so: $dy_p: T_p \mathbb{R}^2\to \mathbb{R}$, i.e. it takes 1-simplexes defined in space ...

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Invariance of volume form under coordinate transformations
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3 votes

Using basic coordinate transformation rules for tensors ($g_{ij}$ and $dx^i$), as well as antisymmetric properties of the wedge product, show that the wedge product $dx^1\wedge\dots dx^n$ transforms ...

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What object exactly is $\frac{d \bar{z}}{dz}$?
2 votes

I think you are looking at a curve specified by: $$z=z\left(t\right) \quad\&\quad \bar{z}=\bar{z}\left(t\right) \quad \mbox{etc.}$$ Where $t$ is the parameter you choose to parametrize your curve ...

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How do I find the bounds for an integral between a plane and a sphere?
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2 votes

Perhaps it would be easier to start by finding the volume of a small slice of the sphere created by leaving only the portion of the sphere that lies in the range $z=a\dots b,\,b>a$. This section ...

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Understanding the proof of non-additivity of outer measure
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1 votes

I think it goes like this. You have established that $\left|V\right|>0$ so you can choose $n>6/\left|V\right|$ such that $\sum_{k=1}^n\left|\left(V+r_k\right)\right|=n\left|V\right|>6$, but ...

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How to calculate the Rotation Matrix from other known values?
1 votes

Firstly, I think the extra gymnastics with $t$ is a distraction, so I will state the problem as: $\mathbf{Q}=\left[\mathbf{q}_1\,\dots,\mathbf{q}_k\right]=\mathbf{P}-\mathbf{t}=\mathbf{R}.\boldsymbol{\...

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How to Compute the Rotation Matrix?
1 votes

Thanks to @David K. I was suspecting that Matlab was doing something convenient, but I don't use it any more, so wanted to avoid guessing. So you have $\mathbf{Q}=\left[\mathbf{q}_1\,\mathbf{q_2}\,\...

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Interesting properties of Green's function
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1 votes

In general, Greens functions do not tend to be well-behaved at all points in space, which can lead to problems. For example consider finding a solution to electromagnetic radiation problem, using the ...

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Proof that $\frac{d}{dt} \log |A(t)| = \text{Tr}\left[A(t)^{-1} \frac{d}{dt} A \right]$
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1 votes

Provided the problem is well-defined, i.e. matrix is diagonalizable, non-singular and determinant is positive, one should be able to use the orthonormality of eigenvectors, i.e. $$ \mathbf{u}^T_i.\...

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Levi-Civta symbol question
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1 votes

Your indexing seems wrong. In the first equation on the left you have three $k$-s, I guess one of them must be $l$. Next, still first line, assuming the second $k$ is $l$: $$ \epsilon_{ijk}\epsilon_{...

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Sphere's surface area element using differential forms
1 votes

As pointed out by @Ted Schifrin, the differential form I have used at the outset was wrong. Here's how I think one can arrive at the correct form. I would appreciate any comments. General approach ...

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Why doesn't ds appear in the statement of Green's Theorem?
1 votes

You seem to be trying to represent a 1-form $\omega=L\left(\mathbf{r}\right) dx + M\left(\mathbf{r}\right) dy$ with a single scalar function $f=f\left(\mathbf{r}\right)$. This will probably not work ...

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How to find the tangent hyperplane in $n$-dimensions?
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0 votes

Line, plane etc. are just words unless you have a good definition in your head, and the problem is that these things can be defined in different ways. As far as I see, there is no single best ...

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Jacobian Matrix and dot product question
0 votes

You don't need to worry about defining Jacobian for this. This is simple Taylor's theorem: $$ f_i\left(y\right)=f_i\left(x+(y-x)\right)=f_i\left(x\right)+\sum_j \partial_j f_i\left(x\right)\,\left(y_j-...

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Eigenvalues of a 3x3 orthogonal matrix
0 votes

I am assuming your matrix is real-valued (since it would not make sense to talk about orthogonality otherwise). Since matrix is orthogonal, it is a normal operator -> it can be diagonalized ($\...

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Prove that $\int_S\left(d\vec\sigma\times \vec\nabla\right)\times \vec P=\int_{\partial S}d\vec r\times \vec P~.$
0 votes

A good way to deal with these things is to learn how to use the Levi-Civita symbols, but since you are asking this question I will assume you are not familiar with it. Another good idea is to simplify ...

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Orthogonal 3D rotational matrices
0 votes

IMHO proving that operator that does rotation in 3D space, is relatively simple - write out the matrix: $\left(\begin{array} \ \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta &...

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Action of a 1-form on the push-forward and pull-back of a vector
0 votes

I think, conventionally the pull-back is defined as adjoint to push-forward. So if you have a manifold $\bar{\mathcal{M}}$, and manifold $\mathcal{M}$ (possibly the same manifold). You then need a map ...

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