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Tuesday, April 14, 2020
Sunday, April 12, 2020
Integration to sum
The definite integral of a continuous function $f$ over the interval $[a,b]$, denoted by $\int_a^b f(x) dx$, is the limit of a Riemann sum as the number of subdivisions approach infinity.
\[ \int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n \delta x \cdot f(x_i) \]
where $\delta x = (b-a)/n$, and $x_i = a+\delta x \cdot i$.
Example
\[\int_{\pi}^{2\pi} cos(x)dx \]
\[\delta x = (b-a)/n = (2\pi-\pi)/n = pi/n\]
\[x_i = a + \delta x \cdot i = \pi + \pi/n \cdot i \]
Therefore
\[\int_{\pi}^{2\pi} \cos(x)dx = \lim_{n\to \infty}\sum_{i=1}^{n} \frac{\pi}{n}\cdot \cos(\pi+\frac{\pi i}{n}) \]
Reference:
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum
\[ \int_a^b f(x)dx = \lim_{n \to \infty} \sum_{i=1}^n \delta x \cdot f(x_i) \]
where $\delta x = (b-a)/n$, and $x_i = a+\delta x \cdot i$.
Example
\[\int_{\pi}^{2\pi} cos(x)dx \]
\[\delta x = (b-a)/n = (2\pi-\pi)/n = pi/n\]
\[x_i = a + \delta x \cdot i = \pi + \pi/n \cdot i \]
Therefore
\[\int_{\pi}^{2\pi} \cos(x)dx = \lim_{n\to \infty}\sum_{i=1}^{n} \frac{\pi}{n}\cdot \cos(\pi+\frac{\pi i}{n}) \]
Reference:
https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-3/a/definite-integral-as-the-limit-of-a-riemann-sum
Monday, April 6, 2020
Thursday, April 2, 2020
Math resources
residue theorem
https://math.mit.edu/~jorloff/18.04/notes/topic8.pdf
A table of integrals of exponential integral - NIST Page
https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf
Ten lessons I should have been taught. Lecture delivered by Gian-Carlo Rota at MIT on April 20, 1996, on occasion of the Rotafest.
Gian-Carlo Rota
USEFUL VECTOR AND TENSOR OPERATIONS
https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1
https://math.mit.edu/~jorloff/18.04/notes/topic8.pdf
A table of integrals of exponential integral - NIST Page
https://nvlpubs.nist.gov/nistpubs/jres/73B/jresv73Bn3p191_A1b.pdf
Ten lessons I should have been taught. Lecture delivered by Gian-Carlo Rota at MIT on April 20, 1996, on occasion of the Rotafest.
Gian-Carlo Rota
USEFUL VECTOR AND TENSOR OPERATIONS
https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118127575.app1
// with very good examples
Riccati equation
\[ dy/dx = a(x) + b(x)y + c(x)y^2 \]
quasi-linear newton Broyden’s Method
Table of Approximations for Derivatives // important for finite difference
SDE
pde
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