1Derivatives and coordinates
IA Vector Calculus
1.2 Inverse functions
Suppose
g, f
:
R
n
→ R
n
are inverse functions, i.e.
g ◦ f
=
f ◦ g
=
id
. Suppose
that f(x) = u and g(u) = x.
Since the derivative of the identity function is the identity matrix (if you
differentiate x wrt to x, you get 1), we must have
M(f ◦ g) = I.
Therefore we know that
M(g) = M(f )
−1
.
We derive this result more formally by noting
∂u
b
∂u
a
= δ
ab
.
So by the chain rule,
∂u
b
∂x
i
∂x
i
∂u
a
= δ
ab
,
i.e. M(f ◦ g) = I.
In the n = 1 case, it is the familiar result that du/dx = 1/(dx/du).
Example.
For
n
= 2, write
u
1
=
ρ
,
u
2
=
ϕ
and let
x
1
=
ρ cos ϕ
and
x
2
=
ρ sin ϕ
. Then the function used to convert between the coordinate systems is
g(u
1
, u
2
) = (u
1
cos u
2
, u
1
sin u
2
)
Then
M(g) =
∂x
1
/∂ρ ∂x
1
/∂ϕ
∂x
2
/∂ρ ∂x
2
/∂ϕ
=
cos ϕ −ρ sin ϕ
sin ϕ ρ cos ϕ
We can invert the relations between (x
1
, x
2
) and (ρ, ϕ) to obtain
ϕ = tan
−1
x
2
x
1
ρ =
q
x
2
1
+ x
2
2
We can calculate
M(f) =
∂ρ/∂x
1
∂ρ/∂x
2
∂ϕ/∂x
1
∂ϕ/∂x
2
= M(g)
−1
.
These matrices are known as Jacobian matrices, and their determinants are
known as the Jacobians.
Note that
det M(f ) det M(g) = 1.