Read Back Signals in Magnetic Recording - Research Group Fidler

Appendix A
The theorem
Appendix A
∫grad f dV = � ∫ f dA
(A.1)
V ∂V
is often used in literature, but it is normally limited to continuously differentiable functions.
Nevertheless this relation is also frivolously applied for functions with first order
singularities, as for example
1
f () r = with V
r−r ′
′∈ r . (A.2)
We were not able to find any proof, which shows the validity of the above theorem for this
function. Thus it is given here by our own.
3
Let us assume any constant vectorc
∈ � and start from the equality
divcf= c grad f . (A.3)
Integration over the whole volume V gives
divcfdV = � cgrad
f dV . (A.4)
∫ ∫
V V
If f has continuous first order partial derivatives we can apply the Divergence Theorem
resulting in
� ∫ c f dA= ∫cgrad
f dV . (A.5)
∂V
V
c is constant and therefore it can be brought in front of the integral.
c� ∫ f dA= c∫grad
f dV . (A.6)
∂V
V
Last equation holds for any c, therefore c can be cancelled, and the theorem (A.1) is verified.
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