# Simple vector calculus

1. Nov 9, 2008

### stunner5000pt

1. The problem statement, all variables and given/known data
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 dependant on the above answer?

2. Nov 9, 2008

### asleight

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

3. Nov 9, 2008

### stunner5000pt

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

4. Nov 10, 2008

### asleight

There is an infinite amount of parallel solutions.