Differentiate equation (12.40) with respect to zand show
∂F
∂z +∂F
∂w∂w∂z = 0 or
∂w
∂z =−∂F
∂z
∂F
∂w(12 .43)provided ∂F∂w = 0.
one equation, n-unknowns
Any equation of the form
F(x 1 , x 2 ,... , x n, w ) = 0 (12 .44)implicitly defines was one or more functions of the n-variables (x 1 , x 2 ,... , x n). It is
left as an exercise to show that for a fixed integer value of jbetween 1 and nthat
∂w
∂x j=−∂F
∂xj
∂F
∂w, provided ∂F
∂w= 0 (12 .45)two equations, three unknowns
Given two equations having the form
F(x, y, z) = 0 and G(x, y, z ) = 0 (12 .46)
then these equations define implicitly (a) zas a function of xand (b) yas a function
of x. Treat z=z(x)and y=y(x)and differentiate each of the equations (12.46) with
respect to xand show
∂F
∂x +∂F
∂ydy
dx +∂F
∂zdz
dx =0
∂G
∂x +∂G
∂ydy
dx +∂G
∂zdz
dx =0(12 .47)The equations (12.47) represent two equations in the two unknowns dydx and dzdx which
can be solved. Use Cramers rule^2 and show these equations have the solutions
dy
dx =−∣∣
∣∣
∣∂F
∂x∂F
∂z
∂G
∂x∂G
∂z∣∣
∣∣
∣
∣∣
∣∣
∣∂F
∂y∂F
∂z
∂G
∂y∂G
∂z∣∣
∣∣
∣and
dz
dx =−∣∣
∣∣
∣∂F
∂y∂F
∂x
∂G
∂y∂G
∂x∣∣
∣∣
∣
∣∣
∣∣
∣∂F
∂y∂F
∂z
∂G
∂y∂G
∂z∣∣
∣∣
∣(12 .48)where ∣
∣∣
∣∣∂F
∂y∂F
∂z
∂G
∂y∂G
∂z∣∣
∣∣
∣= 0(^2) See Cramers rule in Appendex B