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Discovery of the Logarithm by John Napier

Here is a short series of videos I made about the discovery of the Logarithm. John Napier is credited with the discovery of the logarithm, and Leonhard Euler developed the logarithm used today. The contents of the following videos are based on Reference 1. Part 1 Part 2 Part 3 Part 4 References [1] e, The […]

Investigating the Meaning of a Function to the Right of the Del Operator

Let $F(x,y,z) \equiv f(x) g(y) h(z)$. In this previous post, $\vec{\nabla}F(x,y,z)$ was written in terms of spherical coordinates and unit vectors for a spherical coordinate system. The corresponding equation was found to be $ \vec{\nabla}F(x,y,z) = \vec{e}_r \frac{\partial F(x,y,z) }{\partial r} + \vec{e}_\theta \frac{1}{r} \frac{\partial F(x,y,z) }{\partial \theta} + \vec{e}_\phi \frac{1}{r\sin\theta} \frac{\partial F(x,y,z) }{\partial \phi} $. This is (6) […]

Writing Del in Terms of Spherical Coordinates

Objective The objective of this post is to investigate the validity of (6) in Reference 2. In this reference, (6) is del is written in terms of spherical coordinates and spherical unit vectors. Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the this post, the following three equations were written. $ \frac{\partial […]

Writing Unit Vectors for a Cartesian Coordinate System in Terms of Unit Vectors for a Spherical Coordinate System

Objective of this Post The objective of this post is to form (4) in this reference. Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the previous post, the following three equations were written. $ \frac{\partial F(x,y,z)}{\partial x} = \cos\phi \sin\theta \frac{\partial F}{\partial r} + \frac{\cos \phi \cos\theta}{r} \frac{\partial F}{\partial \theta} – […]

How a Multivariable Function Changes with Respect to the Cartesian Coordinates

Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. This post shows how to calculate a partial derivative of a multivariable function $F(x,y,z)$ with respect to each Cartesian coordinates. The resulting expressions are in spherical coordinates. In previous posts (see references [2], [3], and [4]), the following equations were written: $ \frac{\partial }{\partial r} F(x,y,z) = \bigg( ( \frac{\partial x}{\partial r} […]

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 […]

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