Frechet derivative calculation using the differential method

Formulae (9.59) and (9.60) play an important role in electromagnetic inversion. 9.1.7 Frechet derivative calculation using the differential method We now present another way of calculating the Frechet derivative using the differential method, proposed by McGillivray et al., 1994. 

Let us introduce auxiliary electric and magnetic fields, E and H, satisfying the equations 

Multiplying the first equation (9.48) by E and equation (9.62) by H, subtracting one from another, and taking into account the vector identity 

We introduce now a domain Vr, bounded by a sphere dVa of a radius / , with its center at the origin of some Cartesian coordinate system, x, y, z. Integrating both sides of equation (9.65) over the domain Vr, and applying Gauss s theorem, we find  

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