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 acute (i.e. less than 90 degrees) angles in right triangles. In order to proceed with the objectives in this post, it is necessary to extend the allowed values of an angle such that an entire circle can be described if needed. Hence the range of values allowed by $\phi$ is now $[0,360)$. The brackets indicate that the smallest possible angle is $0$ degrees and the largest is less than 360 degrees. It is important to note that several of the trig identities previously derived were not derived with these angular values in mind, so for the time being I hope that either those identities are still valid or that they do not apply to the objectives of the current post. This will be investigated more thoroughly later. 

Suppose that $\phi$ ranging from 0 inclusive to 360 exclusive does not contradict any of the prior conclusions. Then the unit vector $\vec{e}_{\phi}$ should be tangent to a circle in the $x-y$ plane and of any radius, at a certain value of the angle $\phi$.

Now for a difficult question.

What direction does $\vec{e}_{\phi}$ point? Since it is tangent to a circle and perpendicular to a radius, this vector can point either in the direction of increasing $\phi$ or in the direction of decreasing $\phi$. I do not know which to choose! This is quite confusing, how there is an ambiguity in the direction of $\vec{e}_\phi$. Modern convention dictates that $\vec{e}_{\phi}$ should point in the direction of increasing $\phi$, but what is the reason for this convention? I think it may be related to the so-called right-hand-rule, but I am not completely sure. Currently, I am using a Cartesian coordinate system, but out of habit I usually assume that the Cartesian coordinate system is a right-handed coordinate system.

The distinction between left and right handed coordinate systems in Cartesian coordinates simply dictates which direction the $+z$-axis points out of the $x-y$ plane. So, in a sense, there is an ambiguity in the direction of the $+z$-axis in the Cartesian coordinate system. This ambiguity is resolved by distinguishing between right and left handed coordinate systems, in which one of the unit vectors, $\vec{e}_z$, points either up or down with respect to the $x-y$ axes. This is suspiciously analogous to the issue with $\vec{e}_{\phi}$–which of two directions should it point? In the end, the unit vectors of the spherical coordinate system should be perpendicular at a single point in space just like the unit vectors for the Cartesian coordinate system. Assuming there really are only two unique options for the orientations of 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.

In conclusion for today, I need to better understand what a right-handed coordinate system is. Out of context, this is kind of embarrassing to admit. But left- and right- handed coordinate systems will be the topic of the next post. Lots of tangents today.

References:

[1] http://www.thphys.nuim.ie/Notes/MP469/Laplace.pdf

About Author

I am a research assistant in theoretical physics. This website, derive-it.com, serves to organize my ongoing learning and research as well as to provide a resource to other learners around the world.

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