What Is the General Form of Functions in a Divergence-Free Vector Field?

[MathNerd]

V(x,y,z) is a cartesian vector field with components X(x,y,z), Y(x,y,z) and Z(x,y,z) respectively. I am just wondering what is the general form of the functions X, Y and Z as solutions to div( V ) = 0? Where div( V ) is the divergence of the vector field.

Thanks in advance.

 

[Hurkyl]

If I recall correctly, the general solution is

$$\vec{V} = \nabla \times \vec{A} = \mathrm{curl} \vec{A}$$

for any (differentiable) vector field $\vec{A}$.

 

[M98Ranger]

The general form of the functions X, Y, and Z as solutions to div(V) = 0 can be expressed as follows:

X(x,y,z) = f1(y,z)
Y(x,y,z) = f2(x,z)
Z(x,y,z) = f3(x,y)

where f1, f2, and f3 are arbitrary functions of their respective variables.

This form satisfies the condition of zero divergence, as the divergence of a vector field is defined as the sum of the partial derivatives of its components with respect to their corresponding variables. Since the partial derivatives of the functions f1, f2, and f3 with respect to their respective variables are independent of each other, their sum will always be zero.

In other words, this general form of X, Y, and Z ensures that the vector field V(x,y,z) has a constant magnitude in all directions, resulting in a zero divergence.

I hope this helps clarify your understanding of vector differential equations and their solutions.