When I’m sparing some time solving problems, I noticed something I forgot. Including the Derivative of the upper limit of an Integral, the Inertia Index of Quadric Form, etc. Each is basic.

Following the first one, I found a beautiful theorem called the Leibniz Integral Rule, which seems easily overlooked before.

It says

ddx∫a(x)b(x)f(x,t)dt=f(x,b(x))dbdx−f(x,a(x))dadx+∫a(x)b(x)∂∂xf(x,t)dt\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x,b(x)) \frac{db}{dx} - f(x,a(x)) \frac{da}{dx} + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt

when I first saw it I think: why is it the simple summation of all parts? I then got a promising insight that it originates from the Linearity of Integral and Differential operator.

When switched to the second one, I started by learning about the Inertia Index, which is the number of specific eigenvalues.

Lately I found it interesting that the Eigenvalue of Adjacency Matrix is related to its amount of walks with certain length.

Many years ago, I have known about graph theory, but blinded for the elegant part of it. Now more and more wonderful things are revealed to me.

I also looked around the Laplacian Matrix, which is the matrix representation of the graph.

The Laplacian Matrix combines the Adjacency Matrix and the Degree Matrix ingeniously as

L=D−AL = D - A

This two matrices doesn’t interfere mutually, because they have exclusive entries. If you want to know the Degrees of a graph, just look at the diagonal elements of the Laplacian Matrix, or all the “-1” entries for the Adjacency Relation.

In this time, I spot a field terminology called Spectral Graph Theory, maybe it’s the one I’m admiring and talking about.

The Next is a simple review of the Convergence Criteria for Series. I had not thought about the deeper essence of those criterias, but they seem sensible intuitively.

So we have three kinds of convergence criteria for series in usual cases:

  1. The Ratio Test

L=limn→∞∣an+1an∣L = lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|

  1. The Root Test

L=limn→∞∣an∣nL = lim_{n \to \infty} \sqrt[n]{|a_n|}

  1. The Comparison Test

I’ve collected other methods, but forgot in the corner by myself. I’ll find a chance to write them down.

By the way, it’s a pity that Zed’s markdown editing experience is very poor, many vital quick inputs are missing, or else I could write more contents and arrange these more elegantly. I’m learning to get it better, or even contribute an extension for it.