Divergenceless vector function - can we draw component by componet conclusion?

[bjnartowt]

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

Homework Statement

Is this true or false?

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

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

Thanks!

 

[dextercioby]

No, of course not, unless the components of A in the normal basis are numbers independent of the point (x,y,z) in which the vector is defined. In other words, A is a constant vector field.