Simplify curl$(\mathbf{F} \times \mathbf{G})$ under these hypotheses on $F, G$ - 2011 4C

Question

Hypotheses: $F$ is a constant vector
$\nabla \times F = \mathbf{ 0 }$
$\nabla \cdot F = 0 $
$\nabla \cdot G = 0 $

From my other post, $[\color{green}{\nabla} \color{brown}{\times} (\mathbf{F} \times \mathbf{G})]_i = (G \cdot \nabla)\mathbf{F} + F_i(\nabla \centerdot \mathbf{G}) - (\nabla \cdot \mathbf{F})G_i - \color{darkred}{ (\mathbf{F} \cdot \nabla) } G_i $.

Thanks to $\nabla \centerdot G = 0 $, the second term is 0.
Thanks to $\nabla \centerdot F = 0 $, the third term = 0.

So $RHS = (G \cdot \nabla)\mathbf{F} + 0 - 0 - \color{darkred}{ (\mathbf{F} \cdot \nabla) } G_i $

$1.$ The $\cdot$ denotes dot product, so does $\color{darkred}{ \mathbf{F} \cdot \nabla = \nabla \cdot \mathbf{F} }$ ? If so, then the 4th rightmost term would = 0. But the given answer is $ -\color{darkred}{ (\mathbf{F} \cdot \nabla) } G_i $, so why is it not 0?

$2.$ Following the above question, how does the 1st term = 0 ?

$3.$ The question states: $F$ is a constant vector. But shouldn't it write "vector field" ? Or do these identities apply to both vectors and vector fields?

Answer 1

First we have to understand what does $F\cdot \nabla$ means. Remember that $\nabla$ is not a vector of numbers (or functions) but an operator, and accordingly it behaves differently.

When you write $\nabla \cdot F$ it is a shorthand for $\frac {\partial f_1} {\partial x}+\frac {\partial f_2} {\partial y}+\frac {\partial f_3} {\partial z}$. Similarily, $F\cdot \nabla=f_1 \frac \partial {\partial x} +f_2 \frac \partial {\partial y} +f_3 \frac \partial {\partial z} $.

But, unlike real numbers (or functions returning real numbers), $f_1$ and $\frac \partial {\partial x}$ doesn't commute, i.e. $\frac {\partial f_1} {\partial x}$ and $f_1 \frac \partial {\partial x}$ are NOT the same thing. The first is a FUNCTION, the partial derivative of $f_1$ , and the second is an OPERATOR. When apllied to a funcion, it returns its partial derivative wrt x, multiplied by the function $f_1$. That is, $(f_1 \frac \partial {\partial x})g(x)=f_1(x)\frac {\partial g} {\partial x}(x)$.

This should answer your first question. No, they are not the same. $\nabla \cdot F$ is a real function, actually a scalar field. $F\cdot \nabla$ is a differential operator acting on scalar fields.

The answer to the second question is that the differential operator $G\cdot \nabla$ returns zero when acting on $F$ since $F$ is a constant field so any derivative of it is zero.

For the third question, a constant vector is often identified with the constant FUNCTION (or field) that returns that same vector. So $F$ when the question says "$F$ is a constant vector" it actually means that $F$ is the constant field that returns the value $F$.