and
are
functions continuously differentiable on a region
As
ranges
over
the
point
generates
a region
in
the
plane.
University Maths Notes: Calculus – Changes of Variables and Integration – The Jacobian
Suppose thatandare
functions continuously differentiable on a region
Asranges
overthe
pointgenerates
a regionin
theplane.
If the mapping
is
one to one on the interior of
and
the Jacobian
on
the interior of B then the area of
Suppose then that we want to integrate some continuous
function
over
If
the integral is intractable then we can change variables to
and
integrate overinstead,
because
Proof: Break upinto
smaller
regions
We
can write
The last expression is a Riemann sum for
and
the expression tends to this integral as the diameter of the
tends
to zero.