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

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SUMMARY

The discussion centers on the properties of a divergenceless vector function, specifically whether the condition \(\nabla \cdot \mathbf{A} = 0\) implies that each component of the vector field \(\mathbf{A}\) is constant across the domain. The conclusion drawn is that this implication is false unless the components of \(\mathbf{A}\) are constants independent of the spatial coordinates (x, y, z). Therefore, a divergenceless vector field does not necessitate that all its components are zero or constant.

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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!
 
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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.
 

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