## WHAT IS THE BEST PROOF OF CAUCHY’S INTEGRAL THEOREM?

Sourse: Kevin Houston

http://www.kevinhouston.net/blog/2013/03/what-is-the-best-proof-of-cauchys-integral-theorem/

## Môn Toán kỹ thuật ở MIT

Đây là các phần Toán kỹ thuật được dạy ở Massachusetts Institute of Technology (Học viện Công nghệ Massachusetts).

## Course Textbook

Kreyszig, Erwin. Advanced Engineering Mathematics. 8th ed. New York, NY: J.W. Wiley & Sons, 1999. ISBN: 9780471333753.

# Mathematics for Materials Scientists and Engineers

Parabolic approximation to a surface and local eigenframe. The surface on the left is a second-­order approximation of a surface at the point where the coordinate axes are drawn. The surface has a local normal at that point which is related to the cross product of the two tangents of the coordinate curves that cross at the that point. The three directions define a coordinate system. The coordinate system can be translated so that the origin lies at the point where the surface is expanded and rotated so that the normal n coincides with the z-axis as in the right hand curve. (Image by Prof. W. Craig Carter.)

### Instructor(s)

Prof. W. Craig Carter

3.016

Fall 2005

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## Course Description

This course covers the mathematical techniques necessary for understanding of materials science and engineering topics such as energetics, materials structure and symmetry, materials response to applied fields, mechanics and physics of solids and soft materials. The class uses examples from the materials science and engineering core courses (3.012 and 3.014) to introduce mathematical concepts and materials-related problem solving skills. Topics include linear algebra and orthonormal basis, eigenvalues and eigenvectors, quadratic forms, tensor operations, symmetry operations, calculus of several variables, introduction to complex analysis, ordinary and partial differential equations, theory of distributions, and fourier analysis.

Users may find additional or updated materials at Professor Carter’s 3.016 course Web site.

## Bài giảng Toán kỹ thuật của thầy Lê Bá Long

Dear all,

Hiện nay các em đang dùng nhiều giáo trình và bài giảng khác nhau của môn Toán kỹ thuật. Do đó thầy đưa link thư viện lên đây để ta thống nhất dùng Bài giảng của PGS Lê Bá Long, theo version này nhé.

http://dlib.ptit.edu.vn/handle/123456789/1277

Best,

D.

## A expression of sine

Expansion of $\ln(1-z)$

$\sum_{n=1}^{\infty} \frac{z^n}{n}=\ln(1-z)$

Hence, $\sum_{n=1}^{\infty} \frac{\sin(n\theta)}{n} = \sum_{n=1}^{\infty}\frac{e^{in\theta}-e^{-in\theta}}{2in} = \frac{\ln(1-e^{i\theta})-\ln(1-e^{-i\theta})}{2i} = \frac{\pi-\theta}{2}$.

We can apply above formula: $\sum_{n=1}^{\infty} \frac{\sin^3 n}{n} = \sum_{n=1}^{\infty} \frac{3\sin n -\sin(3n)}{4n} = \frac34\cdot\frac{\pi-1}{2} - \frac14\cdot\frac{\pi-3}{2} = \frac{\pi}{4}$.

https://artofproblemsolving.com/community/c7t443f7h1390328_convergence_of_sine_series