Homework Help: Ecuations Resolution

1. Jan 3, 2005

Raparicio

Dear Friends,

How can be resolved a ecuation like this?

$${ \frac{ \partial{(m \vec {v} - \Psi \vec {v})}}{ \partial {t} } = \nabla (\Psi \vec {v}^2 )$$

Asuming that v could be any vector, m a constant, and psi a wave function. It's not the similar that a wave ecuation.

And more:

Is this ok? $$\Psi (\vec {v} \nabla) \vec {v} = \Psi \vec {v} (\nabla \vec {v})$$

2. Jan 3, 2005

dextercioby

So i should understand that the unknown from your equation would be $\Psi$ ??That is to say,all other quantites are known...
So your equation should be looking like that
$$-\vec{v}\frac{\partial\Psi}{\partial t}=\vec{v}^{2}\nabla\Psi+\Psi\nabla(\vec{v}^{2})-m\frac{\partial\vec{v}}{\partial t}$$

I'lm afraind your equation is not 'balanced'.It has only one unknown and three equations.It's actually a system of 3 differential eq.with partial derivatives,but still only one function.Now,if 'v' is an unknown vector function as well,then the eq.(the system of eq.is not 'balanced' again).This time it would 4 unknowns,but only three quations.

My guess,it's something fishy with your eq.Double check it.

Write it in tensor notation.Simplify through the (assumed nonzero) scalar function $\Psi$ and u'll get
$$v_{i}\frac{\partial v_{j}}{\partial x_{i}} \vec{e}_{j}$$
is the LHS.
$$v_{j}\frac{\partial v_{i}}{\partial x_{i}} \vec{e}_{j}$$
is the RHS.
You can see they're different.The first is a contracted tensor product between the vector 'v' and its gradient (which is a second rank tensor),while the second is the product between the vector and its divergence (which is a scalar).Though 'balanced' wrt to tensor rank,the two sides are different.
Therefore,u don't have an equality.

Daniel.

Last edited: Jan 3, 2005