What Happens When You Multiply a Vector by Its Gradient Squared?

[noon0788]

No, this isn't for homework. I'm trying to brush up on gradients for the Physics GRE. Let's say I have vector V.

What is V(gradient squared)V ?

In english, if I multiply the vector to the gradient squared of the same vector, what is that? I'm thinking that it'd just be the gradient of V, but I can't seem to prove it.

Thanks!

 

[HallsofIvy]

Do you mean
\vec{V}\nabla^2\vec{V}? That is commonly called "del squared". "gradient" is specifically \nabla f where f is a scalar valued function. "curl f" is \nabla\times \vec{f} where f is a vector valued function, and "div f" is \nabla\cdot \vec{f}.

Now, \nabla^2= \frac{\partial^2}{\partial x^2}+ \frac{\partial^2}{\partial y^2}+ \frac{\partial }{\partial z^2} (in Cartesian coordinates) is also, strictly speaking, applied to a scalar function but it is not uncommon to see it applied to a vector meaning \nabla^2 applied to each component.

That is, if
\vec{v(x,y,z)}= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k} then

\nabla^2 \vec{v}(x,y,z)= \nabla^2f \vec{i}+ \nabla^2g\vec{j}+ \nabla^2h \vec{k}

a vector. But to say what "\vec{v}\nabla^2\vec{v}" means we would still have to know what kind of vector product that is- dot product or cross product?

If that is dot product and \vec{v}(x,y,z)= f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k} then
\vec{v}\cdot \nabla^2\vec{v}= f(x,y,z)\nabla^2 f(x,y,z)+ g(x,y,z)\nabla^2 g(x,y,z)+ h(x,y,z)\nabla^2h(x, y, z)

If, instead, it is the cross product then
\vec{v}\times\nabla^2\vec{v}= \left(g(x,y,z)\nabla^2h(x,y,z)- h(x,y,z)\nabla^2g(x,y,z)\right)\vec{i}- \left(f(x,y,z)\nabla^2h(x,y,z)- h(x,y,z)\nabla^2f(x,y,z)\right)\vec{i}+ \left(f(x,y,z)\nabla^2g(x,y,z)- g(x,y,z)\nabla^2f(x,y,z)\right)\vec{k}

 

[arildno]

It might also, Halls, be an outer product, i.e a matrix

 

[HallsofIvy]

Good point. In that case it would be
\begin{bmatrix}f(x,y,z)\nabla^2f(x,y,z) & f(x,y,z)\nabla^2g(x,y,z) & f(x,y,z)\nabla^2h(x,y,z) \\ g(x,y,z)\nabla^2f(x,y,z) & g(x,y,z)\nabla^2g(x,y,z) & g(x,y,z)\nabla^2h(x,y,z) \\ h(x,y,z)\nabla^2f(x,y,z) & h(x,y,z)\nabla^2g(x,y,z) & h(x,y,z)\nabla^2h(x,y,z)\end{bmatrix}
in the x, y, z, coordinate system.