three equations, five unknowns
A set of equations having the form
F(x, y, u, v, w ) =0
G(x, y, u, v, w ) =0
H(x, y, u, v, w ) =0(12 .69)implicitly defines u, v, w as functions of xand yso that one can write
u=u(x, y ) v=v(x, y ) w=w(x, y ) (12 .70)The partial derivatives of u, v and w with respect to x and y are calculated in a
manner similar to the previous representations presented.
In order to save space in typesetting sometimes the notation for partial deriva-
tives is shortened to the use of subscripts. For example, one can define
ux=∂u
∂x, u y=∂u
∂y, F x=∂F
∂x, F xx =∂(^2) F
∂x^2
, F w=∂F
∂w
, F xy = ∂
(^2) F
∂x ∂y
, etc
We now employ this notation and take the partial derivatives of equations F, G and
Hwith respect to xand write
Fx+Fuux+Fvvx+Fwwx=0
Gx+Guux+Gvvx+Gwwx=0
Hx+Huux+Hvvx+Hwwx=0(12 .71)The equations (12.71) represent three equations in the three unknowns ux, vxand wx
which can be solved using Cramers rule. This can be accomplished by defining the
3 by 3 Jacobian determinant
∂(F, G, H )
∂(u, v, w )=∣∣
∣∣
∣∣
∣Fu Fv Fw
Gu Gv Gw
Hu Hv Hw∣∣
∣∣
∣∣
∣=∣∣
∣∣
∣∣
∣∂F
∂u∂F
∂v∂F
∂w
∂G
∂u∂G
∂v∂G
∂w
∂H∂u ∂H∂v ∂H∂w∣∣
∣∣
∣∣
∣(12 .72)If this Jacobian determinant is different from zero, the system of equations (12.71)
has a unique solution for ux, vx, w xgiven by
ux=−∂(F,G,H )
∂(x,v,w )
∂(F,G,H )
∂(u,v,w ), vx=−∂(F,G,H )
∂(u,x,w )
∂(F,G,H )
∂(u,v,w ), w x=−∂(F,G,H )
∂(u,v,x)
∂(F,G,H )
∂(u,v,w )(12 .73)