Welcome to my homepage!

This is a dumb undergraduate majoring in physics in an unknownuniversity. I've studied some astronomy myself but haven't figured outanything T_T

But I really want to study astrophysics aaaaaaaaaaaaaaaa

I may post some study materials and daily thoughts here.

This semester there are three courses: Optics, Theoretical Mechanicsand Mathematical Methods for Physics. The only one that's normal is MMP(after all, it's the God of MMP, Guoquan Zhou). But the teachers of theremaining two classes are a bit crazy.

Theoretical Mechanics

The textbook is Landau, which is not suitable for beginners. Thisbook is not suitable as a textbook for beginners, it has no mathematicaldetails and the notation is different from other books. In the end, Ichose to follow along with other books.

And I'm a sophomore, sophomore, sophomore, and I haven't finished themath and physics stuff yet, and even though you're in astrophysics andI'm interested in it, all of a sudden there's a lot of metrics andgeneral relativity and abstract metrics and tensor analysis and it justmakes me feel like I'm taking a very abstract class.

Conclusion: self-study.

Optics

I don't have a problem with the teacher's lectures, but can byd go inorder of difficulty and generalization. The first lesson first byd talkMaxwell's system of equations how to derive the electromagnetic waveequation, where I learned vector analysis uh uh uh, have to talk aboutthe introduction to electrodynamics is it. Then the second class is aquick lesson in geometric optics, not even polarization, just interfacereflection and refraction Fresnel formula, I byd 8 am already sleepy andstill have to give me a sleep aid, right?

Conclusion: self-study.

Conclusion

College is about developing your self-learning skills, and so are therequired courses.🤣👉

The key lies in the fact that Prof. Gao combines some of our existingconcepts with some advanced mathematical tools, and introduces newconcepts in a more concise and easy-to-understand language.

Generalization of thePythagorean Theorem

We know that in flat Euclidian space, the distance between two pointscan be expressed by the Pythagorean theorem: \[\begin{equation}\label{eq1}\mathrm{d}s^2=\mathrm{d}x^2+\mathrm{d}y^2,\end{equation}\] or in polar coordinates: \[\begin{equation}\label{eq2}\mathrm{d}s^2=\mathrm{d}\rho^2+\rho^2\mathrm{d}\varphi^2.\end{equation}\]

So in other spaces, the Pythagorean theorem still holds as long asone takes infinitely small distances so that the spaceis approximately flat and linear. For example, on a sphere: \[\begin{equation}\label{eq3}\mathrm{d}s^2=R^2\mathrm{d}\theta^2+R^2\sin^2\theta\,\mathrm{d}\varphi^2.\end{equation}\]

Write thePythagorean theorem in quadratic form

Although the Pythagorean theorem holds in different spaces andcoordinate systems, their forms seem to be less uniform (e.g. \(\eqref{eq1}\), \(\eqref{eq2}\) and \(\eqref{eq3}\)). But each of them is asquare term. This prompted us to think of the quadratic forms we learnedin linear algebra. So we try to write the above three Pythagoreantheorems in the form of quadratic matrices: \(\eqref{eq3}\) and \(\eqref{eq3}\). \[\begin{align}\mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}x&\mathrm{d}y\end{pmatrix}\begin{pmatrix}1&0\\0&1\end{pmatrix}\begin{pmatrix}\mathrm{d}x\\\mathrm{d}y\end{pmatrix},\\[12pt]\mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}\rho&\mathrm{d}\varphi\end{pmatrix}\begin{pmatrix}1&0\\0&\rho^2\end{pmatrix}\begin{pmatrix}\mathrm{d}\rho\\\mathrm{d}\varphi\end{pmatrix},\\[12pt]\mathrm{d}s^2&=\begin{pmatrix}\mathrm{d}\theta&\mathrm{d}\varphi\end{pmatrix}\begin{pmatrix}R^2&0\\0&R^2\sin^2\theta\end{pmatrix}\begin{pmatrix}\mathrm{d}\theta\\\mathrm{d}\varphi\end{pmatrix}.\end{align}\] It turns out that \[\begin{equation}\mathrm{d}s^2\equiv\mathrm{d}\boldsymbol{\rho}^\mathrm{T}\,G\,\mathrm{d}\boldsymbol{\rho}.\end{equation}\] Uniform in form, now that's a good thing. We can write ageneralized form \[\begin{equation}\mathrm{d}s^2=\begin{pmatrix}\mathrm{d}\rho^1&\cdots&\mathrm{d}\rho^s\end{pmatrix}\begin{pmatrix}g_{11}&\cdots&g_{1s}\\\vdots&\ddots&\vdots\\g_{s1}&\cdots&g_{ss}\end{pmatrix}\begin{pmatrix}\mathrm{d}\rho^1\\\vdots\\\mathrm{d}q^s\end{pmatrix}.\end{equation}\] So we define the matrix \(g_{ab}\) as a metric. Sothe above three examples and other spatial/targeting metrics are clear:P.

So to summarize, the metric is a generalization of thePythagorean theorem to infinitesimal distances.

Final roundproblems of Chinese National Astronomy Olympiad(2002-2020)

Final round problemsof Chinese National Astronomy Olympiad (2021-2022)

Video forProblem 3 in CNAO 2021-2022

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