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**divergenceless vector function - can we draw "component by componet" conclusion?**

## Homework Statement

Is this true or false?

[tex]\nabla \bullet {\bf{A}} = \frac{{\partial {A_i}}}{{\partial {x_i}}} + \frac{{\partial {A_j}}}{{\partial {x_j}}} + \frac{{\partial {A_k}}}{{\partial {x_k}}} = 0{\rm{ }} \to {\rm{ }}\frac{{\partial {A_i}}}{{\partial {x_i}}} = \frac{{\partial {A_j}}}{{\partial {x_j}}} = \frac{{\partial {A_k}}}{{\partial {x_k}}} = 0[/tex]

...in which the arrow says "implies that".

Thanks!