Defining Del in Terms of Cartesian Coordinates

What is Del?

In math, the symbol $\vec{\nabla}$ is called “del.” This symbol is defined in terms of Cartesian coordinates.

$\vec{\nabla} := \vec{e}_x\frac{\partial}{\partial x} + \vec{e}_y\frac{\partial}{\partial y} + \vec{e}_z\frac{\partial}{\partial z}$

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 $\nabla$.In the above definition, the unit vectors are placed to the left of the derivatives, to emphasize that the derivatives do not act on the Cartesian unit vectors.

The $:=$ symbol indicates that $\vec{\nabla} $ is defined in terms of the right side, rather than the right side defined in terms of the left side. The reason for making this distinction is that $\vec{\nabla}$ is a new quantity that should be defined in terms of known quantities, instead of known quantities being defined in terms of a new quantity, which is a little strange because the known quantities have already been defined. In my opinion, it does not make sense to define a quantity more than once.

Note that partial derivatives are used on the right side.

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:

$\vec{\nabla} := \vec{e}_x\frac{\partial}{\partial x} + \vec{e}_y\frac{\partial}{\partial y} + \vec{e}_z\frac{\partial}{\partial z}$.

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 is related to an ambiguity concerning 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 $x-y$ plane. This is suspiciously analogous to the issue with $\vec{e}_{\phi}$–which of two directions should it point?

The unit vectors of the spherical coordinate system are perpendicular at a single point in space just like the unit vectors for the Cartesian coordinate system. 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.

In conclusion for today, I need to better understand what a right-handed coordinate system is.

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