The symbol $\nabla$ represents the vector operator del: $$\nabla = \left\langle {\partial \over \partial x},\; {\partial \over \partial y}, \;{\partial \over \partial z}\right\rangle.$$ To use a more economical (if less standard) notation, this can also be written as $$\nabla = \left\langle \partial_x,\;\partial_y,\;\partial_z\right\rangle,$$ wherein $\partial_q$ denotes ${\partial \over \partial q}$ for an arbitrary coordinate $q.$

Although vector operators don’t have magnitudes and directions like ordinary vectors do, $\nabla$ can do just about anything a vector can do, including scalar multiplication, dot products, and cross products. Since taking derivatives of constants always gives us zero, using $\nabla$ on individual vectors like $\langle 2, -5, 6 \rangle$ isn’t all that interesting. Instead, we’ll be using it on scalar functions and vector fields, the latter of which are functions $\vec v(x,y,z)$ that assign a vector to every point in 3D space. (The official definition is much more general, but for E&M we’ll be almost exclusively interested in vector fields over $\mathbb{R}^3.)$

Acting on a scalar function $f$ with the operator $\nabla$ creates a vector function called the gradient of $f$: $$\nabla f = \left\langle {\partial \over \partial x}, \;{\partial \over \partial y}, \;{\partial \over \partial z}\right\rangle f = \left\langle {\partial f \over \partial x}, \;{\partial f \over \partial y}, \;{\partial f \over \partial z}\right\rangle.$$ Like $f$ itself, $\nabla f$ takes the coordinates of a point in 3D space as its input. The output is a vector whose components are the partial derivatives of $f$ with respect to the three coordinate axes, whose direction indicates the direction of steepest increase for $f,$ and whose magnitude indicates the size of that increase. (In other words: the rate of change of $f,$ in the direction parallel to $\nabla f,$ is equal to $|\nabla f|.$)

(One comment on notation: a lot of people add subscripts to a function to denote its partial derivative with respect to that coordinate. For example, they might use $f_x$ to mean ${\partial f \over \partial x}.$ There’s nothing wrong with this notation, but since we’ll be working so much with vector functions in this course, and vector functions have components which are regularly denoted using subscripts, I’ll reserve the subscript notation $v_x$ to mean “the $x$-component of $\vec v$”, and use either ${\partial f \over \partial x}$ or $\partial_x f$ to mean “the derivative of $f$ with respect to $x$”.)

We can also calculate the rate of change of $f$ in the direction of an arbitrary unit vector $\hat u.$ This is called the directional derivative of $f$ (in the direction of $\hat u),$ and it is calculated as the dot product between the gradient and the direction vector: $$D_{\hat u}f = \nabla f \bullet \hat u.$$ According to the formula above for the magnitude of a dot product, we can also write this as $D_{\hat u}f = |\nabla f|\cos\alpha,$ where $\alpha$ is the angle between $\nabla f$ and $\hat u.$ (Since $\hat u$ is a unit vector, the standard factor of $|\hat u|$ simply multiplies the expression by $1.)$ Due to the range of output values for cosine, this shows us that the derivative of $f$ is largest when $\alpha = 0$ (i.e. when $\hat u$ points in the same direction as $\nabla f$), smallest when $\alpha = \pi$ (i.e. when $\hat u$ points opposite the direction of $\nabla f$), and zero when $\alpha = \tfrac\pi2$ (i.e. in any direction perpendicular to the direction of $\nabla f$).

Further additions:

• divergence (net flow through a particular point, flow out minus flow in)
• curl (both the magnitude and direction of the cross product are meaningful, which is a theme that will later resurface when we discuss surface integrals)
• summary table of inputs and outputs for these different uses of del (e.g. gradient, input scalar function, output $\nabla f = \langle \partial_x f, \partial_y f, \partial_z f \rangle$ (vector) )