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 $$\newcommand{pdv}[2]{\frac{\partial #1}{\partial #2}} \shaded{ \mathrm{curl}\left(\vec F\right)=\pdv{N}{x}-\pdv{M}{y}=N_x-M_y } \nonumber$$ 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.
Quadratic equations Derives the equation for the roots of a general quadratic equation.
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.
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
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,
Math Talk Describes the hardware schematic for connecting a FPGA and an Arduino. Implements the protocol to transfer bytes and the messages using the Verilog HDL.
Building Math Hardware This article shows some implementation of the math operations introduced in chapter 7 of the inquiry "How do computers do Math?" The combinational logic is described in the HDL Verilog 2001.
Continuous Transfer Functions Transfer function in the Laplace domain let us determine the system response characteristics without having to solve the underlying differential equation.
Impedance Derives the formula for impedance of common passive electronic components using the models for energy storage of those parts.
Laplace in Mechanical Systems Analyzes springs, resistance and mass in mechanical systems using Laplace transforms to solve the ordinary differential equations.
Laplace transform Proof of Laplace transforms as used in my Electronics article
Euler’s Formula Proofs Euler's formula using the MacLaurin series for sine and cosine. Introduces Euler's identify and Cartesian and Polar coordinates.
How do computers do math? Detailed explanation of semi-conductor chemistry, semi-conductors physics, to logic gates elementary and arithmetic operations.