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This question already has an answer here:
I've been explained that a vector field, when seen as "arrows" in the plane, has 0 divergence when its magnitude doesn't change, i.e. when the "arrows" keep same length. But the following examples puzzle me: $F(x)=x/|x|$ has always norm 1 but its divergence is not 0 $F(x)=x/|x|^2$ has not constant norm but its divergence is 0 Is there some contradiction or do I have a wrong/incomplete picture? |
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I don't think your examples contradict your view on the divergence. Take your first example, the divergence is $$\frac{\partial}{\partial x}\left(\frac{x}{\sqrt{x^2}}\right) = \frac{\sqrt{x^2} - x\frac{2x}{2\sqrt{x^2}}}{x^2} = \frac{\sqrt{x^2} - \frac{x^2}{\sqrt{x^2}}}{x^2} = 0.$$ And similarly we find for your second example: $$\frac{\partial}{\partial x}\left(\frac{x}{x^2}\right) = \frac{x^2 - x\cdot2x}{x^4} = -\frac{x^2}{x^4} = -\frac{1}{x^2},$$ which is clearly not zero. |
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