The same pattern persists as in the 2-dimensional case (see Key Point 10). Across a row of the determinant the numerators are the same and down a column the denominators are the same.
The volume element
becomes
. As before the limits and integrand must also be transformed.
The change of coordinates from Cartesian to spherical polar coordinates is given by the transformation equations
We now need the nine partial derivatives
Hence we have
Check that this gives
. Notice that
for
, so
. The limits are found as follows. The variable
is related to ‘latitude’ with
representing the ‘North Pole’ with
representing the equator and
representing the ‘South Pole’.
The variable
is related to ‘longitude’ with values of
to
covering every point for each value of
. Thus limits on
are
to
and limits on
are
to
. The limits on
are
(centre) to
(surface).
To find the volume of the sphere we then integrate the volume element
between these limits.
This will be a difficult integral to derive limits for in terms of
,
and
. However, it can be noted that the base is described by
while the upper face is described by
. Similarly, the front face is described by
with the back face being described by
. Finally the left face satisfies
while the right face satisfies
.
The above suggests a change of variable with the new variables satisfying
,
and
and the limits on
being
to
, the limits on
being
to
and the limits on
being
to
.
Inverting the relationship between
,
,
and
,
and
, gives
The Jacobian is given by
Note that the function
equals
. Thus the integral is