So, I have to verify the product rule by direct calculation, and I know the product rule for curl is $\operatorname{grad}(fg)=f(\operatorname{grad}(g)) + g(\operatorname{grad}(f))$.
Sorry, I don't know how to format it correctly. Is there a defined way to use direct calculation to verify that this is true? I've also never seen the product rule notation as listed above before, so can someone please explain it to me. Thanks, and much appreciated!
If you just crunch it out by components you can see that it works out. $$\begin{align}\vec\nabla\times(f\vec G)&=\left\langle\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right\rangle\times\left\langle fG_x,fG_y,fG_z\right\rangle\\ &=\left\langle\frac{\partial(fG_z)}{\partial y}-\frac{\partial(fG_y)}{\partial z},\frac{\partial(fG_x)}{\partial z}-\frac{\partial(fG_z)}{\partial x},\frac{\partial(fG_y)}{\partial x}-\frac{\partial(fG_x)}{\partial y}\right\rangle\\ &=\left\langle\frac{\partial f}{\partial y}G_z-\frac{\partial f}{\partial z}G_y,\frac{\partial f}{\partial z}G_x-\frac{\partial f}{\partial x}G_z,\frac{\partial f}{\partial x}G_y-\frac{\partial f}{\partial y}G_x\right\rangle\\ &\quad+f\left\langle\frac{\partial G_z}{\partial y}-\frac{\partial G_y}{\partial z},\frac{\partial G_x}{\partial z}-\frac{\partial G_z}{\partial x},\frac{\partial G_y}{\partial x}-\frac{\partial G_x}{\partial y}\right\rangle\\ &=\left\langle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\right\rangle\times\left\langle G_x,G_y,G_z\right\rangle+f\left\langle\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right\rangle\times\left\langle G_x,G_y,G_z\right\rangle\\ &=\left(\vec\nabla f\right)\times\vec G+f\vec\nabla\times\vec G\end{align}$$ I'm not sure about the product rule notation you are referring to is. Can you clarify a bit?
