[Physics] Why are divergence and curl related to dot and cross product

Question

I've been reading Griffith's intro to electrodynamics and I've been a bit confused about his explanation of divergence and curl. I don't understand how divergence is the dot product of a gradient acting on a vector function and curl is the cross product of gradient acting on a vector function. Does it relate to the fact that one uses sine while the other uses cosine? Just to clarify, I understand the concept of divergence and curl from a purely conceptual standpoint, it's just this mathematical definition that I can't wrap my head around.

Answer 1

First of all let's define dot product and cross product between two 3-vectors $$\mathbf{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} \qquad \text{and} \qquad \mathbf{b} = \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} $$

dot product: $$ \mathbf{a}\cdot\mathbf{b} = \sum_i a_i b_i = a_1 b_1 + a_2 b_2+ a_3b_3 $$

cross product: $$ \mathbf{a}\times\mathbf{b} = \begin{pmatrix} a_2 b_3 - a_3 b_2 \\a_3 b_1 - a_1 b_3 \\a_1 b_2 - a_2 b_1 \end{pmatrix} $$

Note that these definitions do not involve geometric quantities like the angle between the two vectors; indeed, it is the angle that is defined in terms of the dot product (for the records, $\cos \theta := \mathbf{a\cdot b}/ \sqrt{\mathbf{(a\cdot a)(b\cdot b)}}$).

Then you have the definition of divergence and curl acting on a function $\mathbf{f}(\mathbf{x}) \equiv \begin{pmatrix}f_1(\mathbf{x}), f_2(\mathbf{x}), f_3(\mathbf{x})\end{pmatrix}$ ($\mathbf{x} = (x_1,x_2,x_3)$; you can call $x_1=x$, $x_2=y$ and $x_3=z$ but my choice allow a compact notation):

divergence: $$ \mathrm{div}\, \mathbf{f} := \frac{\partial }{\partial x_1} f_1+\frac{\partial }{\partial x_2} f_2+\frac{\partial }{\partial x_3} f_3 = \sum_i \frac{\partial }{\partial x_i}f_i \equiv \sum_i {\partial_i}f_i $$ where $\partial_i \equiv \partial / \partial x_i$.

curl: $$ \mathrm{curl} \,\mathbf{f} := \begin{pmatrix} \partial_2 f_3 - {\partial_3 f_2} \\ \partial_3 f_1 - \partial_1 f_3 \\ \partial_1 f_2 - \partial_2 F_1\end{pmatrix} $$

Now you can see that if you introduce the quantity $$ \nabla = \begin{pmatrix} \partial_1 \\ \partial_2 \\ \partial_3 \end{pmatrix} $$ you can write the operations of divergence and curl as if $\nabla$ was a vector! Indeed if you apply the definition of dot and cross product you can easily find out that $$ \nabla \cdot \mathbf{f} = \mathrm{div}\, \mathbf{f} \qquad \text{and} \qquad \nabla \times \mathbf{f} = \mathrm{curl}\, \mathbf{f} $$ You can find out that many identities holding for 3-vectors still hold id one of them is $\nabla$.

But note that this "trick" of thinking to $\nabla$ as a 3-vector is formal and not all identities holding for usual 3-vectors keep working.