Question

Why divergence of curl is zero?

Answer 1

damn calc 3 lol

Answer 2

When you carry out the vector operations, yu get either zeros due to the orthogonality of certain vectors or you get zeros die to differences of identical quantities. Carry it out with div curl r or some-such.

Answer 3

It's a fairly straightforward proof, but messy in its implementation.

Suppose F(x, y, z) = (Fx(x, y, z), Fy(x, y, z), Fz(x, y, z)) is a vector field with twice-continuously-differentiable components.

Then by the definition of curl in rectangular coordinates:

curl F = (dFz/dy - dFy/dz, dFx/dz - dFz/dx, dFy/dx - dFx/dy)

Where "d" represents the "di" of partial differentiation.

No apply a div to this vector field:

div curl F = d/dx(dFz/dy - dFy/dz) + d/dy(dFx/dz - dFz/dx) + d/dz(dFy/dx - dFx/dy)
=d^2Fz/dxdy - d^2Fy/dxdz + d^2Fx/dydz - d^2Fz/dydx + d^2Fy/dzdx - d^2Fx/dzdy

Rearranging the terms then gives:

div curl F = (d^2Fz/dxdy - d^2Fz/dydx) + (d^2Fy/dzdx - d^2Fy/dxdz) + (d^2Fx/dydz - d^2Fx/dzdy)

Since the components of F are sufficiently smooth, we can reverse the order of partial differentiation with impunity, which means that each braketed term above cancels out. So we get.

div curl F = 0

Since F was general (up to a differentiability condition that is likely to be true in all "physical" vector fields) this is true for all vector fields.

Answer 4

Simply it may be written as:

Answer 5

Let u be an arbitrary vector with components f(x,y,z), g(x,y,z), and h(x,y,z) in the i, j, and k directions respectively.

Thus
u = < f(x,y,z) , g(x,y,z) , h(x,y,z) >

Calculate the curl of u ( curl(u) = del x u ). The x stands for the cross product.
curl(u) = < d/dx , d/dy , d/dz > x < f , g , h >
curl( u) = < dh/dy - dg/dz , df/dz - dh/dx , dg/dx - dg/dy >

Caluculate the divergence of the curl ( div( curl(u) ) = del * u). The * stands for dot product.
div( curl(u) ) = < d/dx , d/dy , d/dz > * < dh/dy - dg/dz , df/dz - dh/dx , dg/dx - df/dy >

div( curl(u) ) = d^2h / dxdy - d^2g / dxdz + d^2f / dydz - d^2h / dydx + d^2g / dzdx - d^2f/ dzdy.

Because of the equality of mixed partial derivatives,
d^2h / dxdy = d^2h / dydx
d^2g / dxdz = d^2g / dzdx
d^2f / dydz = d^2f / dzdy

all of the terms on the right hand side cancel out and you're left with

div( curl(u) ) = 0.

Good Luck!