I'm trying to figure out what $\nabla(F\cdot G)$ would be, given F and G are vector-valued functions. I know that $F \cdot G$ is the scalar product, so $\nabla (F \cdot G)$ must give a vector-value of some sort. I have found two formulae, which I'm fairly certain are equal but the dimensions of one don't seem to add up.
The first is $\nabla (F\cdot G) = (F \cdot \nabla)G + (G \cdot \nabla) F + F \times (\nabla \times G) + G \times (\nabla \times F)$. I know that each of these quantities are vectors, so $\nabla (F \cdot G)$ must also be a vector, and my assumption is correct.
Issues come in with the second one, $\nabla (F \cdot G) = F \cdot \nabla G + G \cdot \nabla F$. I know that the gradient of a scalar-valued function is a vector, and the gradient of a vector-valued function is the Jacobian of that vector, but I don't think the dimensions match up here? The closest things that I could see which would make this legal (and I'm certain I'm getting something wrong here) one of two things.
First, if we have $\nabla (F \cdot G) = (\nabla F)G + (\nabla G)F$, this would at least allow the dimensions to match but this isn't what's said?
Second, we use the definition of the dot product, and write $F \cdot \nabla G = F^{T}\nabla G$, which again the dimensions would allow, and this is what I suspect it is but I just want to double check before I assume something and make an even sillier mistake down the line!
My question is, how does this second formula make sense? I'm pretty certain I'm being silly and missing something obvious, but unfortunately I can't see what! Any help would be greatly appreciated, thank you!
Best Answer
Let's expand. $$\nabla(F\cdot G) = F \cdot \nabla G + G \cdot \nabla F$$ Let $G = (G_x, G_y, G_z)$ and $\nabla G = (\nabla G_x, \nabla G_y, \nabla G_z) $. To take the dot product of that with $F$, we would take $(F\cdot \nabla G_x, F\cdot \nabla G_y, F \cdot \nabla G_z) $
Meanwhile, $G \cdot \nabla F = (G \cdot \nabla F_x, G \cdot \nabla F_y, G \cdot \nabla F_z)$. Each entry is a scalar, so together that is a vector. The sum of both is $$((F\cdot \nabla G_x + G \cdot \nabla F_x,F\cdot \nabla G_y+ G \cdot \nabla F_y, F \cdot \nabla G_z + G \cdot \nabla F_z)$$
That looks like a vector made of three scalar expressions to me.