Finding Y Component for Vector Field F with Zero Divergence

[stunner5000pt]

Homework Statement

Suppose we have the vector field F whose x component is given by $F_{x}=Ax$ and whose divergence is known to be zero $\vec{\nabla}\cdot\vec{F}=0$, then find a possible y component for this field. How many y components are possible?

2. The attempt at a solution

So the divergence in cartesian coordinates is given by
$$\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} = 0$$

Using the fact that $$F_{x}=Ax$$
$$A+\frac{\partial F}{\partial y} = 0$$
$$\frac{\partial F}{\partial y} = -A$$
integrate both sides with respect to y we get

$$F_{y}=-Ay+B$$

where B is a constant
is that sufficient for a possible y component? For the question with howm any are possible... arent there infinite possibilities since B could be anything. But they are all parallel to each... linearly dependent on the above answer?

 

[asleight]

Homework Statement

Suppose we have the vector field F whose x component is given by $F_{x}=Ax$ and whose divergence is known to be zero $\vec{\nabla}\cdot\vec{F}=0$, then find a possible y component for this field. How many y components are possible?

2. The attempt at a solution

So the divergence in cartesian coordinates is given by
$$\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} = 0$$

Using the fact that $$F_{x}=Ax$$
$$A+\frac{\partial F}{\partial y} = 0$$
$$\frac{\partial F}{\partial y} = -A$$
integrate both sides with respect to y we get

$$F_{y}=-Ay+B$$

where B is a constant
is that sufficient for a possible y component? For the question with howm any are possible... arent there infinite possibilities since B could be anything. But they are all parallel to each... linearly dependent on the above answer?

As far as I'm concerned, you're good to go.

 

[stunner5000pt]

As far as I'm concerned, you're good to go.

but.. infinitely many solutions because of B or finite because they are all linearly dependent on the solution given?

 

[asleight]

but.. infinitely many solutions because of B or finite because they are all linearly dependent on the solution given?

There is an infinite amount of parallel solutions.