Flux in a vector field My notes of“Denis Auroux. 18.02 Multivariable Calculus. Fall 2007. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.” Please watch these excellent lectures, and then come back for a reference. Flux in a plane Definition Let $$C$$ be a plane curve, and $$\vec F$$ be a vector field in that plane. Then the …
Curl of a vector field My notes of“Denis Auroux. 18.02 Multivariable Calculus. Fall 2007. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.” Please watch these excellent lectures, and then come back if you need a reference. Definition $$\newcommand{pdv}{\frac{\partial #1}{\partial #2}} \shaded{ \mathrm{curl}\left(\vec F\right)=\pdv{N}{x}-\pdv{M}{y}=N_x-M_y } \nonumber$$ For a gradient field $$N_x=M_y$$, so the $$\mathrm{curl}(\vec F)=0$$. The …
Vectors These are my notes of:These are my notes of the lectures by Denis Auroux. 18.02 Multivariable Calculus. Fall 2007. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA. Description will use a plane $$\mathbb{R}^2$$, or space $$\mathbb{R}^3$$, but the same principles apply to higher dimensions. Vectors are commonly displayed on the $$xyz$$-axis, with …
Line Integration Vector calculus is about differentiation and integration of vector fields. This article gives an overview of the integration operations.
Curve length Length of the arc in function graphs and parametric functions.
Google Assistant switches Sonoff S20 Switch your light on/off using your voice and the help of Google Assistant. Sure, you can run to the store and purchase a preconfigured light switch, but what's the fun in that and more importantly these switches with their close-source software require access to your home network.
Our vegetarian recipes Large collection of our proven vegetarian recipes. Some original, some adjusted to fit our personal taste and ingredients available.
Scalable IoT Integration (ESP32+Google IoT) We're only using the IoT device and Google Cloud Platform. The only programming will be in JavaScript (nodejs). No opening ports on the firewall, no IFT3. Plain and simple, 'though never as simple as getting up and flipping a light switch, or opening the outside door to see how the weather is.
DD-WRT Reverse Proxy and HTTPS (Asus RT-AC68, Pound, LetsEncrypt) Describes how to use DD-WRT as a Reverse Proxy with HTTPS. It relies on pound for the reverse proxy and LetsEncrypt for the TLS certificate. This configuration was tested on an Asus RT-AC68, but should also work on other routes with DD-WRT firmware.
Complex Functions All our arithmetic functions gracefully extend from the one-dimensional number line into the 2-dimensional $$\mathbb{C}$$-plane. We will introduce the functions that operate on complex arguments.
Complex Numbers Instead of projecting the future merits of complex numbers, we will introduce them in an intuitive way. We draw a parallel to negative numbers that have been universally accepted around the same time.
RL High-pass Filter Trigonometry method, example 2 Assume the non-homogeneous lineair differential equation of a first order High-pass LC-filter, where $$u(t)=\hat{u}\cos(\omega t)$$ is the forcing function and the current $$i(t)$$ through the inductor is the response. The differential equation for this system is $$L\,{i_p}^\prime(t)+R\,i_p(t)=\hat{u}\cos(\omega t)\label{eq:bTrigRL_DV}$$ The solution is a superposition of the natural response and a …
Laurent Series Named after Pierre Alphonse Laurent, a French mathematician and Military Officer, published in the series 1843. The Laurent series is a representation of a complex function f(z) as a series. Unlike the Taylor series which expresses $$f(z)$$ as a series of terms with non-negative powers of $$z$$, a Laurent series includes terms with negative powers. …
Z-transform Proofs Proofs for Z-transform properties, pairs, initial and final value. Includes derivative, binomial scaled, sine and other functions.
Partial Fraction Expansion Oliver Heaviside (1850-1925), was an English electrical engineer, mathematician and physicist who among many things adapted complex numbers to the study of electrical circuits. He introduced a method to decompose rational function of polynomials as they occur when using the Laplace transform to solve differential equations. Whenever the denominator of a rational function can be …
Binomial Theorem Proof of Isaac Newton generalized binomial theorem
LFZ Transform references stanford, 2 ref SA chapter 33 from The Scientist & Engineer’s Guide to Digital Signal Processing dspcan http://www.emba.uvm.edu/~gmirchan/classes/EE275/Handouts_Ed4/Ch06(4e)Handouts/Ch6(1)Handouts_4e.pdf http://www.emba.uvm.edu/~gmirchan/classes/EE275/Handouts_Ed4/Ch06(4e)Handouts/Ch6(2)Handouts_4e.pdf matlab https://www.math.uci.edu/icamp/courses/math77a/lecture_10f/rational_filters.pdf single pole filter frequency response https://ccrma.stanford.edu/~jos/fp/Elementary_Filter_Sections.html https://ccrma.stanford.edu/~jos/filters/filters.html see https://ccrma.stanford.edu/~jos/filters/Stability_Revisited.html http://fourier.eng.hmc.edu/e102/lectures/Z_Transform/node2.html http://www.eecg.toronto.edu/~ahouse/mirror/engi7824/course_notes_7824_part6.pdf https://en.wikipedia.org/wiki/Linear_time-invariant_theory http://pilot.cnxproject.org/content/collection/col10064/latest
Inverse Z-transform The inverse Z-transform, can be evaluated using Cauchy's integral. Which is an integral taken over a counter-clockwise closed contour C in the region of converge of (z)
Discrete Transfer Functions Transfer function in the Z-domain let us determine the discrete system response characteristics without having to solve the underlying equations.
Geometric series Derives formulas for finite and infinite geometric power series.
Complex Arithmetic Formulas Complex arithmetic formulas written in LaTex used in the HP-41 program. Includes everything from power to trigonometric functions.
Z-transform Derives the Z-transform using the Laplace transform. Includes stability criteria and region of convergence where the z-transform is valid.
Fourier Transform The Fourier transform takes a time-domain signal and decomposes into the frequencies that make it up,
Mosquito Flat to Gem Lakes (CA) Elevation gain, distance and pictures from a day hike from Mosquito Flat to Gem Lakes in the High Sierra Nevada mountains.
North Lake to Piute Lake (CA) Elevation gain, distance and pictures from a day hike from North Lake to Piute Lake (CA) in the High Sierra Nevada mountains.