A coordinate system can be defined by three perpendicular unit vectors. If the coordinate system is Cartesian, which direction does the $+x$ axis point? To resolve this problem, I define an orientation–a coordinate system that is oriented in a certain direction in three-dimensional space. How many unique orientations are there? Here, a unique orientation is an orientation that cannot be rotated by less than 90 degrees into another orientation —admittedly, 90 degrees is an arbitrary number, but at least it is a start. Two orientations related by less than a quarter of a rotation are considered equivalent in contrast to unique. […]

# Month: February 2021

What is Del? In math, the symbol $\vec{\nabla}$ is called “del.” This symbol is defined in terms of Cartesian coordinates. $\vec{\nabla} \equiv \frac{d}{dx}\vec{e}_x + \frac{d}{dy}\vec{e}_y + \frac{d}{dz}\vec{e}_z$ The right side is a sum of unit vectors. So $\vec{\nabla}$ is a vector. This is why I write $\vec{\nabla}$ instead of just $\nabla$. Is it possible to express $\vec{\nabla}$ in terms of spherical coordinates? I will find out. This post lists nine derivatives relating spherical coordinates to Cartesian coordinates. As suggested in Reference [1], these nine derivatives are useful for expressing del in terms of spherical coordinates. I return to the definition

Review of Integration Integration with Cartesian coordinates is simple. The general form is $\int\int\int f(x,y,z)dxdydz$ in which $f(x,y,z)$ is an arbitrary function of the Cartesian coordinates. However, there may be cases in which integrating with spherical coordinates is more convenient. Given the above, general form for integration with Cartesian coordinates, how can one integrate in a spherical coordinate system? How the Spherical Coordinate System is Different The first step is to convert $f(x,y,z)$ into a function $g(r,\phi,\theta)$ of the spherical coordinates, $r$, $\phi$, and $\theta$. Here, $\phi$ is the azimuthal angle in the $x-y$ plane, and $\theta$ is the polar angle.

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Using conclusions from previous posts, the following nine derivatives have been determined. $\frac{dx(r)}{dr}\big|_{r^+}=\sin\theta\cos\phi$ $\frac{dy(r)}{dr}\big|_{r^+} = \sin\theta\sin\phi$ $\frac{dz(r)}{dr}\big|_{r^+} = \cos\theta$ $\frac{dx(\phi)}{d\phi}\big|_{\phi^+} = -r\sin\theta\sin\phi$ $\frac{dy(\phi)}{d\phi}\big|_{\phi^+} = r \sin\theta \cos\phi$ $\frac{dz(\phi)}{d\phi}\big|_{\phi^+} = 0$ $\frac{dx(\theta)}{d\theta} \big|_{\theta^+} = r\cos\theta \cos\phi$ $\frac{d y(\theta)}{d \theta} \big|_{\theta^+} = r \cos\theta \sin\phi $ $\frac{dz(\theta)}{d\theta} \big|_{\theta^+} = -r \sin\theta$ Next, recall the following result from this post. This matrix equation can be rewritten by substituting the nine derivatives listed at the beginning of the current post. Replacing each $a$ with the corresponding independent variable, the result is: This matrix equation includes nine equations relating dummy variables in Cartesian coordinates to

I would like to evaluate two more derivatives. They are $ \frac{dx}{d\phi}\big|_{\phi^+} $ and $ \frac{dz}{d\theta}\big|_{\theta^+} $ given $x = r \sin\theta\cos\phi$ and $z = r\cos\theta$. Start with $ \frac{dx}{d\phi}\big|_{\phi^+} $. The first step is to substitute $x$ with $r \sin\theta \cos\phi$. $ \frac{dx}{d\phi}\big|_{\phi^+} = \frac{d r \sin\theta \cos \phi}{d\phi}\big|_{\phi^+} $. The next step is to apply the product rule to the right side. $\frac{dx}{d\phi}\big|_{\phi^+} = r \sin\theta \frac{d \cos \phi}{d\phi}\big|_{\phi^+} + \cos \phi \frac{dr \sin\theta}{d\phi}\big|_{\phi^+}$. Using this derivative of cosine, the previous equation becomes $\frac{dx}{d\phi}\big|_{\phi^+} = -r \sin\theta \sin\phi + \cos \phi \frac{d r \sin\theta}{d\phi}\big|_{\phi^+}$. From this post about the derivative

In this post, the derivative of the cosine function is found. To do this, the steps in reference 1 are followed. Start with a definition of a derivative, from this post: $\frac{df(x)}{dx}\bigg|_{a^+} \equiv \lim_{\Delta x \rightarrow 0^+} \frac{ f(a + \Delta x) – f(a) }{\Delta x} $. Since $f(x)$ and $\cos(x)$ are both functions of $x$, replace $f$ with $\cos$: $\frac{d\cos(x)}{dx}\bigg|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \frac{ \cos(a + \Delta x) – \cos(a) }{\Delta x} $. Recall the angle addition identities. Specifically, $ \cos(A+B) = \cos A \cos B – \sin A \sin B$. Using this identity, the numerator of the

In this post, the chain rule is proved. This rule frequently appears in Calculus. Recall from this post that: $dx|_{a^+} \equiv \lim_{\Delta x \rightarrow 0^+} \Delta x$ and $df(x)|_{a^+} = \lim_{\Delta x \rightarrow 0^+} \Delta f (\Delta x)$. Suppose a variable $y$ can be written as a function of another variable $u$, and that $u$ can be written as a function of another variable $x$. Then $y$ can be written as a function of $x$: $y(x)$. What is $\frac{dy(x)}{dx}\bigg|_{a^+}$ if only $y(u)$ and $u(x)$ are known? Solution: Start by writing $\frac{dy(x)}{dx}\bigg|_{a^+} = \frac{dy(x)}{dx}\bigg|_{a^+}$. According to this post, the $|_{a^+}$ symbols can be moved