Del is More than an Upside Down Triangle

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 of del: $\vec{\nabla} \equiv \frac{d}{dx}\vec{e}_x + \frac{d}{dy}\vec{e}_y + \frac{d}{dz}\vec{e}_z$. The first issue I address is expressing the unit vectors $\vec{e}_x$, $\vec{e}_y$, and $\vec{e}_z$ in terms of unit vectors for the spherical coordinate system, since if the del operator is to be represented in terms of spherical coordinates, the unit vectors should be those for spherical coordinates as well; it wouldn’t make much sense to mix Cartesian unit vectors and spherical coordinates! In summary, perhaps it is possible to express $\vec{e}_x$, $\vec{e}_y$, and $\vec{e}_z$ in terms of the spherical coordinate unit vectors, $\vec{e}_r$, $\vec{e}_{\phi}$, and $\vec{e}_{\theta}$. Defining the Radial Unit Vector The unit vector $\vec{e}_r$ has not been precisely defined on this blog yet. In order for $\vec{e}_r$ to point in the correct direction according to the spherical coordinate system, $\vec{e}_r$ must be defined as $\vec{e}_r \equiv \frac{\vec{r}}{r}$. Here, $\vec{r} \equiv x\vec{e}_x + y\vec{e}_y + z\vec{e}_z$ and $r = \sqrt{x^2 + y^2 + z^2}$ from the Pythagorean theorem. Therefore, $\boxed{ \vec{e}_r = \frac{x\vec{e}_x + y\vec{e}_y + z\vec{e}_z}{\sqrt{x^2 + y^2 + z^2}} }$. That was fairly simple. Thoughts on the Azimuthal Unit Vector What is $\vec{e}_{\phi}$ in Cartesian coordinates? Up until this point in this entire blog, angles have been restricted to the values allowed by […]