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…

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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.$…

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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…

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The Radial Unit Vector

In this post, I find an expression for the radial unit vector, $\vec{e}_r$. The three unit vectors in the following digram form a right-handed spherical coordinate system. This unit vector is easier to…

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