• # Notes on String Theory (1)

In this note we'll discuss classical bosonic strings, and briefly introduce the quantization procedure.

This note follows Superstring Theory by Edward Witten, and Introduction to String Theory and M-Theory by Michio Kaku.

• # Curvature 2-forms and tetrad method

In differential geometry, and especially in general relativity, the Riemann curvature tensor is an important object for the geometric property of a manifold, solving the Enstein's equation one always needs to calculate the curvature tensor. But the component expression of Riemann tensor is too complicated, making it painful to calculate it by hand, and less powerful when dealing with various problems. Instead, using curvature 2-form would somehow be better.

• # Long time no see!

As you may have noticed, my site haven't been updated for almost 3 years! This could results from that I have been busy with a lot things, such as exams, final thesis, plus that I'm running out of ideas. But now, I'm in a relatively free period and I want to keep my blog from being abandonded.

My future blogs will mainly be my notes that explain my understanding of things I have learned.

• Belows are old posts

# 有趣的共形变换

维度数学漫步第6集讲了一种很有意思的图片变换：把一张照片放在复平面上，然后对所有的复数都做一个变换$%$并把这个复数对应的点移动到变换后的位置上，得到一张新的图片。新图看起来就像是原图经过一些扭曲后得到的，虽然它整体上看是“跑形”了但我们仍然可以辨认出图上的内容，比如原图是两个人的合影，变换后你也看得出是两个人的合影，只不过两人都被夸张地扭曲了。这样的变换称为共形变换，或全纯变换

为什么这种变换会有这样的性质呢？这是因为它是保角的，即一个角在变换前后的大小不变（也许变换后角的边变成了曲线，这时它的大小就该是这两条曲线在顶点处的切线的夹角）。所以，共形变换也叫保角变换

我在这里写了一个共形变换的程序，大家可以用图片去试着变换一下。

• # 分形艺术：Julia集合与Mandelbrot集合

数学中的分形可以说是一种艺术，我们可以通过分形来构造出各种优美的图片，它具有无限精细的结构，可以被无限放大，并且在这些精细结构中还存在缩小了的整个分形！这就是分形的自相似性。当我在维度数学漫步第5集看到Julia集合Mandelbrot集合时就被它的神奇震撼了，一个简单的变换居然可以创造出如此复杂的分形！下面我们就来讨论一下这个数学上的艺术品吧。

## 重点内容

• Julia和Mandelbrot集合的定义
• 颜色方案
• 自相似结构
• 四维Mandelbrot集合
• 在线计算Julia和Mandelbrot集合（点这里