The determinants in the equations (12.48) are called Jacobian determinants of F
and Gand are often expressed using the shorthand notation
∂(F, G )
∂(y, z) =
∣∣
∣∣
∣
∂F∂y ∂F∂z
∂G
∂y
∂G
∂z
∣∣
∣∣
∣=
∂F
∂y
∂G
∂z −
∂G
∂y
∂F
∂z (12 .49)
In terms of Jacobian determinants the derivatives represented by the equations
(12.48) can be represented
dy
dx =−
∂(F,G )
∂(x,z)
∂(F,G )
∂(y,z)
and dxdz =−
∂(F,G )
∂(y,x )
∂(F,G )
∂(y,z)
(12 .50)
where ∂(F, G )
∂(y, z)
= 0.Note the patterns associated with the partial derivatives and the
Jacobian determinants. We will make use of these patterns to calculate derivatives
directly from the given transformation equations in later presentations and examples.
two equations, four unknowns
Two equations of the form
F(x, y, u, v ) =0
G(x, y, u, v ) =0
(12 .51)
implicitly define uand vas functions of xand yso that one can write u=u(x, y )and
v=v(x, y ). The derivatives of the equations(12.51) with respect to x can then be
expressed
∂F
∂x
+∂F
∂u
∂u
∂x
+∂F
∂v
∂v
∂x
=0
∂G
∂x
+∂G
∂u
∂u
∂x
+∂G
∂v
∂v
∂x
=0
(12 .54)
The equations (12.54) represent two equations in the two unknowns ∂u∂x and ∂v∂x .One
can use Cramers rule to solve this system of equations and obtain the solutions
∂u
∂x =−
∂(F,G )
∂(x,v)
∂(F,G )
∂(u,v)
and
∂v
∂x =−
∂(F,G )
∂(u,x)
∂(F,G )
∂(u,v)
(12 .53)
This solution is valid provided the Jacobian determinant ∂(F, G )
∂(u, v )
is different from
zero.
