The derivation, which closely follows Margenau and Murphy, is done in a generalised coordinate system and later transformed to spherical polar.
 
 

The Generalised System:
 
 
 

Coordinates
Infinitismal displacements
Volume element
Displacement vector

In the Spherical Polar System, we have,
 
 

q₁ = r q₂ = qq₃ = f
 
 

dV = r²sinqdq df

Working in the generalised system,
 
 

Let us evaluate the first two terms (the others will follow similarly)
 
 

The second term is equal to 
 
 

The first term needs more work:
 
 

--Verify the above result by writing the triple products in determinantal form.
 
 

We will now evaluate 
 
 

(Note that )
 
 

Therefore,

The L.H.S. is a curl of a gradient, and therefore, equal to zero, which leads to,
 
 

Giving,

In the above equation, one can cyclically permute the subscripts. Substituting the results in Eqn (3) gives,

Substituting in Eqn (2),
 
 

Combining the above result with Eqn (1),

Reverting to the spherical polar coordinate system, we have,
 
 

h₁ = 1, h₂ = r, h₃ = rsinq, so that,

and finally,

(Check carefully and find some errors !)