The differential operator

A function is a mapping from the elements of one set to another. The most popular functions map generally from a set of numbers to another. For example, the function $f : \mathbb{R} \to \mathbb{R}$ where $f(x) = x^2$ maps a real number to the square of that real number.

But a set is simply a collection of distinct objects, and nothing limits those objects to numbers alone. As wild as the idea may sound, one can define a function that maps to another function. We call this an operator.

If you have done basic calculus, you are already well acquainted with this notion. Consider the following mappings:

$$f(x) = x^2 \quad \longmapsto \quad f'(x) = 2x$$

$$g(x) = \sin(x) \quad \longmapsto \quad g'(x) = \cos(x)$$

$$h(x) = e^x \quad \longmapsto \quad h'(x) = e^x$$

Each of these is a function being sent to another function, its derivative. Something is acting on the entire function, not just on a single number.

Which special operator do you think could perfectly encapsulate this relationship?

The differential operator $\dfrac{d}{dx}$.

When we apply this to multivariable functions, a single $\dfrac{d}{dx}$ is no longer adequate, since there are now multiple variables to differentiate with respect to. It is more suitable to consider the partial differential operator $\dfrac{\partial}{\partial x_i}$, which differentiates with respect to one chosen variable while holding all others fixed.

For instance, take $f(x, y) = x^3 y + \sin(y)$:

$$\frac{\partial f}{\partial x} = 3x^2 y \qquad \text{(} y \text{ is treated as a constant)}$$

$$\frac{\partial f}{\partial y} = x^3 + \cos(y) \qquad \text{(} x \text{ is treated as a constant)}$$

The operator $\dfrac{\partial}{\partial x}$ acts on $f$ and returns a new function. Only the $x$-dependence is differentiated, and everything else is carried through unchanged.

With the machinery we have explained, the nabla operator can be thought of as a collection of partial differential operators. Place those operators into this special vector $\nabla$, and you will obtain something both useful and peculiar.

We define how this vector interacts in three contexts:

1. When it is applied to a scalar function to produce the gradient (link)
2. When it is dotted with a vector field to define the divergence
3. When we take the cross product of $\nabla$ with a vector field, which gives the definition of the curl