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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} –…
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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.$ These are definite integrals because the upper and lower limits of integration are finite numbers. Symmetry of the Sine and Cosine Functions…
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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 radius $r=1$, the circumference of the circle is $2 \pi r = 2 \pi$. Even though $2 \pi$ is a number, it…
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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. For a description of this diagram, refer to the following YouTube video. The Radial Unit Vector in Terms of Spherical Coordinates Suppose $r=1$. Using vector addition, $\vec{r} =…