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

In summary, the conversation discusses the concept of a divergenceless vector function and whether it can be concluded "component by component" based on the equation \nabla \bullet {\bf{A}} = 0. The conclusion is that this is only true if the components of A are constants independent of the point in which the vector is defined.
  • #1
bjnartowt
284
3
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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  • #2


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.
 

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

1. What is a divergenceless vector function?

A divergenceless vector function is a vector field in which the divergence, or the measure of the rate of change of the field at a given point, is equal to zero everywhere. This means that the flow of the field is equal in and out of any given point, resulting in a net flow of zero.

2. How can we determine if a vector function is divergenceless?

To determine if a vector function is divergenceless, we can use the divergence theorem, which states that the flux (or flow) of a vector field through a closed surface is equal to the volume integral of the divergence of the field. If the result is equal to zero, then the vector function is divergenceless.

3. Why is it important for a vector function to be divergenceless?

A divergenceless vector function is important in many areas of physics, such as fluid dynamics and electromagnetism. In fluid dynamics, a divergenceless vector field represents a flow that is incompressible, meaning that the volume of fluid remains constant. In electromagnetism, a divergenceless vector field represents a charge-free region, which allows for the calculation of electric potential and magnetic fields.

4. Can we draw conclusions about the individual components of a divergenceless vector function?

No, we cannot draw conclusions about the individual components of a divergenceless vector function. This is because the divergenceless property applies to the vector field as a whole, not to its individual components. Additionally, the individual components may vary in magnitude and direction, but as long as the net divergence is zero, the vector function is considered divergenceless.

5. How is the concept of a divergenceless vector function used in real-world applications?

The concept of a divergenceless vector function is used in many real-world applications, such as weather forecasting, fluid dynamics in engineering, and the study of magnetic fields in physics. By understanding and utilizing the divergenceless property, scientists and engineers can make accurate predictions and calculations for various systems and phenomena.

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