Introduction In this post, I find an expression for the gradient of a function, in terms of spherical coordinates. This is a continuation of previous posts, such as this one. This post has a lot of symbols, but there is a lot of repetition. Formalism Recall that $ \vec{e}_r = \frac{\partial x(r)}{\partial r}\big|_{r^+} \vec{e}_x + \frac{\partial y(r)}{\partial r}\big|_{r^+} \vec{e}_y + \frac{\partial z(r)}{\partial r}\big|_{r^+} \vec{e}_z $ $ \vec{e}_{\theta} = \frac{1}{r} \frac{\partial x(\theta)}{\partial\theta} \big|_{\theta^+} \vec{e}_x + \frac{1}{r} \frac{\partial y(\theta)}{\partial \theta} \big|_{\theta^+} \vec{e}_y + \frac{1}{r} \frac{\partial z(\theta)}{\partial\theta} \big|_{\theta^+} \vec{e}_z $ $\vec{e}_{\phi} = \frac{1}{r \sin\theta} \frac{\partial x(\phi)}{\partial\phi}\big|_{\phi^+} \vec{e}_x + \frac{1}{r \sin\theta} \frac{\partial y(\phi)}{\partial\phi}\big|_{\phi^+} \vec{e}_{y}$. Multiply the first equation by $\partial r|_{r^+}$, multiply the second equation by […]
In this post, I relate coefficients of unit vectors to derivatives and to a Jacobian matrix that was used in a previous post. Unit Vectors Three unit vectors for a right-handed spherical coordinate system are $ \vec{e}_r = \sin \theta \cos\phi \vec{e}_x + \sin \theta \sin\phi \vec{e}_y + \cos \theta \vec{e}_z $ $ \vec{e}_{\theta} = \cos\theta \cos\phi \vec{e}_x + \cos\theta \sin\phi \vec{e}_y -\sin\theta \vec{e}_z $ $\vec{e}_{\phi} = -\sin\phi \vec{e}_x + \cos\phi \vec{e}_{y}$. In this coordinate system, a point is represented by $(r, \theta, \phi)$. Derivatives Recall the following nine derivatives from previous posts. $\frac{dx(r)}{dr}\big|_{r^+}=\sin\theta\cos\phi$ $\frac{dy(r)}{dr}\big|_{r^+} = \sin\theta\sin\phi$ $\frac{dz(r)}{dr}\big|_{r^+} = \cos\theta$ $\frac{dx(\theta)}{d\theta}
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 find than the other two unit vectors because all that is needed is vector addition. The Radial Unit Vector in Terms of Spherical Coordinates Suppose $r=1$. Using vector addition, $\vec{r} = r’ \cos\phi \vec{e}_x + r’ \sin\phi \vec{e}_y + \cos \theta \vec{e}_z$. Since $r=1$, the expression on the right is equal to $\vec{e}_r$: $\vec{e}_r = r’ \cos\phi \vec{e}_x + r’ \sin\phi \vec{e}_y + \cos \theta \vec{e}_z$.
Consider a spherical coordinate system. Let a point be represented by $(r, \theta, \phi)$, in that order. Now that the order of the coordinates is established, I can define unit vectors that form a right-handed coordinate system. Suppose the radial unit vector $\vec{e}_r$ points radially outward from the origin to the point, and the polar unit vector $\vec{e}_{\theta}$ points in the direction of increasing $\theta$. From the right hand rule, the azimuthal unit vector $\vec{e}_{\phi}$ points in the direction of increasing $\phi$. Below is a diagram with an arbitrary vector $\vec{r}=(r,\theta,\phi)=r \vec{e}_r$. The Polar Unit Vector in Terms of
In this post, I write the azimuthal unit vector $\vec{e}_{\phi}$ in terms of Cartesian coordinates. Here, $\phi$ is the azimuthal angle in the $x-y$ plane. As noted in this post, $\vec{e}_{\phi}$ points in the direction of increasing $\phi$. Geometrical Setup Since $\vec{e}_{\phi}$ is perpendicular to the line segment from the origin to the point $(x,y,0)$, I am looking for a unit vector that is perpendicular to this line segment. A unit vector has a magnitude and a direction, and these properties of the unit vector do not change if I translate the unit vector to the origin. So, I am looking
This post introduced the following questions. What direction does $\vec{e}_{\phi}$ point? Modern convention dictates that $\vec{e}_{\phi}$ should point in the direction of increasing $\phi$, but what is the reason for this convention? If there are only two unique configurations for three mutually perpendicular unit vectors in light of the ambiguous orientation of each unit vector, it seems likely that the direction of $\vec{e}_{\phi}$ dictates whether the unit vectors in spherical coordinates form a right or left handed coordinate system in the same sense that the Cartesian unit vectors do. Unique Configurations First of all, are there only two unique configurations for three
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.
Reasoning about Left and Right Handed Coordinate SystemsRead More »