How to read complicated mathematical equations

Math is nothing more than a concise shorthand for describing relationships between objects.

数学的描述是对生活中对象的简洁表达， 需要联系生活。

Reading a technical book is fundamentally different from reading a novel. The first goal is to understand it. The second goal is to decide whether to keep reading or to pick up another book and get more details so you can figure out what it really means. The end result is that you often bounce from one book to another. Unlike a mystery novel, the goal isn't to blast through the book to see how it ends; the goal is to learn stuff.

这段话也就是在表达：阅读数学书籍，首先要理解，然后理解，然后再去理解，是一个深入理解的过程！

The method I use involves breaking the equation down into chunks and visualizing what the relationship between variables looks like. There are six steps:

1. Decide which of the terms are constants and which contain variables.

确定公式中的常量与变量；

2. Ignore the constants. They're just decoration that will scale the result up or down. Usually the text will tell you what they mean. Physicists often get tired of writing all the constants over and over, so they just set them all to 1.

更加注重变量之间的联系，先忽略公式中的常量；

3. Break each term down and visualize what it would look like when plotted on an x-y graph. There are some things that crop up over and over in formulas. These are easy to memorize. (Mathematicians sometimes claim they never memorize, but they're lying.)
   1. Geometric curves. A term with only constants is just a scale factor or something that shifts the curve in some direction. A term with only one variable and some constants is just a straight line. An x² term would look like a parabola. And so on. Your task is to figure out whether the curve goes up or down, and whether it passes through the origin.
   2. Exponentials. e^x is a curve going up, e^−x is a curve going down.
   3. Sine waves. Sometimes sines, cosines, etc. are spelled out explicitly, but more often they're written with imaginary numbers, that is, with ‘i’. An equation with an e^−ix, for example, is usually not an exponential but a sine wave.
   4. Bessel, Laplace, Gaussian, Airy functions, dot and cross product. These have character­istic shapes or actions that you should memorize. They are the alphabet of math.
   5. If the term is in the denominator, flip the curve vertically in your mind.
   6. Integrals. Take the sum of all the stuff behind the integration sign. For instance, if it's a squiggly curve, you'd visualize a curve inching up or down.
   7. Differentials. These you wouldn't normally visualize (though you could); all you need to know is that instead of the variable, the equation is telling you what happens when the variable changes.

• 第三点是要求对整个公式的每项建立一个可视化的理解：这需要对平时常常接触的数学公式的曲线有所记忆；理解方程中的变量变化时会发生什么。

4. Add or subtract the terms as the equation says, and visualize what would happen to the curve.

• 更深入理解公式的曲线变化，比如加减一项变成什么；

5. Make note of special types of variables. For example, it might be a vector or matrix. If it's a vector, the equation is telling you how its length and direction change. With matrices and tensors, visualizing it is a little tougher: you have to do it in three dimensions.

对特殊类型的变量做记录；如变量、矩阵、张量的维度信息等；

6. Go back and look at the constants to see how it scales.

这时候再去看常量做了什么样的伸缩；