Mathematical Physics Vol 1

4.6 Examples

179

If c is the boundary of this quadrangle, then I c A · d r = Z EF A · d r + Z FG A · d r + Z GH

A · d r + Z HE

A · d r .

The values of these integrals are Z EF

A · d r = A x − A · d r = − A x + A · d r = A y + 1 2 A · d r = − A y − 1 2

∆ y ∆ x

∂ A x ∂ y

∆ y ∆ x

Z GH Z FG Z HE

∂ A x ∂ y

1 2

∆ x ∆ y

∂ A y ∂ x

1 2 ∆ x ∆ y . where all members of order higher than ∆ x , that is, ∆ y , are ignored, ∂ A y ∂ x

Figure 4.32

It follows from here that I c

A · d r =

∂ y

∂ A y

∂ A x

∆ x ∆ y .

∂ x −

As ∆ S = ∆ x ∆ y , the z projection of rot A is

( rot A ) · k = lim ∆ S → 0 H S

A · d r ∆ S ∂ x −

=

∂ A y

∂ y

∂ A x

∆ x ∆ y

= lim ∆ y → 0 ∆ x → 0

=

∆ x ∆ y

∂ A y

∂ A x ∂ y

∂ x −

=

.