d2pdu2[u_,v_]=G111[u,v]*dpdu[u,v]+G112[u,v]*dpdv[u,v]+l[u,v]vi[u,v]

d2pdv2[u_,v_]=G221[u,v]*dpdu[u,v]+G222[u,v]*dpdv[u,v]+n[u,v]vi[u,v]

d2pdudv[u_,v_]=G121[u,v]*dpdu[u,v]+G122[u,v]*dpdv[u,v]+m[u,v]vi[u,v]

dvidu[u_,v_]=a11[u,v]*dpdu[u,v]+a21[u,v]*dpdv[u,v]

dvidv[u_,v_]=a12[u,v]*dpdu[u,v]+a22[u,v]*dpdv[u,v]

detIFF[u_,v_]=e[u,v]*g[u,v]-f[u,v]^2

a11[u_,v_]=(m[u,v]*f[u,v]-l[u,v]*g[u,v])/detIFF[u,v]

a12[u_,v_]=(n[u,v]*f[u,v]-m[u,v]*g[u,v])/detIFF[u,v]

a21[u_,v_]=(l[u,v]*f[u,v]-m[u,v]*e[u,v])/detIFF[u,v]

a22[u_,v_]=(m[u,v]*f[u,v]-n[u,v]*e[u,v])/detIFF[u,v]

podstaw={D[dpdu[u,v],u]->d2pdu2[u,v],D[dpdu[u,v],v]->d2pdudv[u,v],D[dpdv[u,v],u]->d2pdudv[u,v],D[dpdv[u,v],v]->d2pdv2[u,v],D[vi[u,v],u]->dvidu[u,v],D[vi[u,v],v]->dvidv[u,v]}

Schpuuv[u_,v_]=FullSimplify[(D[d2pdu2[u,v],v]-D[d2pdudv[u,v],u])/.podstaw]

puuvdpdv[u_,v_]=FullSimplify[Coefficient[Schpuuv[u,v],dpdv[u,v]]]

numpuuvdpdv[u_,v_]=FullSimplify[Numerator[puuvdpdv[u,v]]]

FullSimplify[Coefficient[numpuuvdpdv[u,v],f[u,v]^2]+Coefficient[numpuuvdpdv[u,v],e[u,v]*g[u,v]]]

row[u_,v_]=Coefficient[numpuuvdpdv[u,v],f[u,v]^2](((f[u,v])^2)-e[u,v]*g[u,v])+e[u,v](-((m[u,v])^2)+l[u,v] n[u,v])

FullSimplify[row[u,v]-numpuuvdpdv[u,v]]

frow[u_,v_]=FullSimplify[row[u,v]/Denominator[puuvdpdv[u,v]]]/.{(( (((m[u,v])^2)-l[u,v] n[u,v]))/(-((f[u,v])^2)+e[u,v] g[u,v]))->-K[u,v]}

FullSimplify[Solve[frow[u,v]==0,K[u,v]]]