## How a Multivariable Function Changes with Respect to an Azimuthal Coordinate

For a functions $f(x)$, $g(y)$ and $h(z)$, the chain rule yields $ \frac{\partial f(x)}{\partial \phi} = ( \frac{\partial f}{\partial x} ) ( \frac{\partial x}{\partial \phi} )$. $ \frac{\partial g(y)}{\partial \phi} = (\frac{\partial g}{\partial y}) (\frac{\partial y}{\partial \phi} )$. $ \frac{\partial h(z)}{\partial \phi} = (\frac{\partial h}{\partial z})( \frac{\partial z}{\partial \phi} )$. Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. This is […]

## How a Multivariable Function Changes with Respect to a Polar Coordinate

For a functions $f(x)$, $g(y)$ and $h(z)$, the chain rule yields $ \frac{\partial f(x)}{\partial \theta} = ( \frac{\partial f}{\partial x} ) ( \frac{\partial x}{\partial \theta} )$. $ \frac{\partial g(y)}{\partial \theta} = (\frac{\partial g}{\partial y}) (\frac{\partial y}{\partial \theta} )$. $ \frac{\partial h(z)}{\partial \theta} = (\frac{\partial h}{\partial z})( \frac{\partial z}{\partial \theta} )$. Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. This is […]

## How a Multivariable Function Changes with Respect to a Radial Coordinate

First focus on the Cartesian coordinate $x$, which depends on the spherical coordinates $r,\theta,$ and $\phi$. For a function $f(x)$, the chain rule yields $ \frac{\partial f(x)}{\partial r} = ( \frac{\partial f}{\partial x} ) ( \frac{\partial x}{\partial r} )$. Similarly for functions $g(y)$ and $h(z)$: $ \frac{\partial g(y)}{\partial r} = (\frac{\partial g}{\partial y}) (\frac{\partial y}{\partial r} […]

## Indefinite Integration

The following quote from Reference [1] can clarify a lot of confusion about indefinite integration: Today the process of finding the fluent of a given fluxion is called indefinite integration, or antidifferentiation, and the result of integrating a given function is its indefinite integral, or antiderivative (the “indefinite” refers to the existence of the arbitrary […]

## Partial Fractions

This post is based on a section called Integration by Partial Fractions in Morris Kline’s book on Calculus [1]. The first step in understanding partial fractions is learning about polynomials. Definition of a Polynomial From Wikipedia, the following statement defines a polynomial [2]. In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and […]

## Short Overview of Special Relativity based on Zangwill

Here is a summary of Section 22.1 (Special Relativity) of Zangwill’s Electrodynamics textbook [1]. Basic Points of Special Relativity There are different observers. One observer moves with a constant velocity with respect to the other observer. A velocity vector has a direction & a magnitude, so a constant velocity means that the moving observer’s direction & […]

## Multiple Cycles of the Complex Exponential Function

A Definite Integral of the Complex Exponential Function Recall from this post that: $ \int_0^{2 \pi} d\theta \cos\theta + i \int_0^{2 \pi} d\theta \sin\theta = 0$ and $ \int_0^{2 \pi} d\theta e^{i \theta} = 0.$ These are definite integrals because the upper and lower limits of integration are finite numbers. Symmetry of the Sine and Cosine Functions […]

## One Cycle of a Trigonometric Function

Integral of a Complex Exponential Consider the integral $ \int_0^{2 \pi} d\theta e^{i \theta} $. Upper Limit of $2 \pi$ Radians The upper limit is the number, $2 \pi$. For a circle with radius $r=1$, the circumference of the circle is $2 \pi r = 2 \pi$. Even though $2 \pi$ is a number, it […]

## Partnership with Myntor

Derive It is glad to announce a new partnership with the startup company, Myntor (www.myntor.io). They are a venture backed edtech startup with 1200+ students that provides AI-powered test prep courses for AP subjects and the SAT/ACT. Students of Myntor’s test preparation programs have been accepted to universities such as MIT and Harvard. The […]

## Equations of Mathematical Physics, Notes on Set Theory

This is a summary of Chapter 1, Section 1 of Reference [1], which is Equations of Mathematical Physics by V.S. Vladimirov. The majority of this post consists of definitions from V.S. Vladimirov. Terms in Set Theory Foundation $A$ is a set. $a \in A$ means element $a$ is contained in set $A$. $B$ is another […]

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