- #1
stunner5000pt
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- 2
Homework Statement
Suppose we have the vector field F whose x component is given by [itex]F_{x}=Ax[/itex] and whose divergence is known to be zero [itex] \vec{\nabla}\cdot\vec{F}=0[/itex], 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
[tex]\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} = 0[/tex]
Using the fact that [tex]F_{x}=Ax[/tex]
[tex]A+\frac{\partial F}{\partial y} = 0[/tex]
[tex]\frac{\partial F}{\partial y} = -A[/tex]
integrate both sides with respect to y we get
[tex]F_{y}=-Ay+B[/tex]
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?