:: deftheorem Def15 defines hpartdiff23 PDIFF_5:def 15 :
for f being PartFunc of (REAL 3),REAL
for u being Element of REAL 3 st f is_hpartial_differentiable`23_in u holds
for b₃ being Real holds
( b₃ = hpartdiff23 (f,u) iff ex x0, y0, z0 being Real st
( u = <*x0,y0,z0*> & ex N being Neighbourhood of z0 st
( N c= dom (SVF1 (3,(pdiff1 (f,2)),u)) & ex L being LinearFunc ex R being RestFunc st
( b₃ = L . 1 & ( for z being Real st z in N holds
((SVF1 (3,(pdiff1 (f,2)),u)) . z) - ((SVF1 (3,(pdiff1 (f,2)),u)) . z0) = (L . (z - z0)) + (R . (z - z0)) ) ) ) ) );