3 The Jacobian in 3 dimensions
When changing the coordinate system of a triple integral
we need to extend the above definition of the Jacobian to 3 dimensions.
Key Point 12
Jacobian in Three Variables
For given transformations , and the Jacobian is
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.
Example 26
Use spherical coordinates to find the volume of a sphere of radius .
Figure 32
Solution
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.
Example 27
Find the volume integral of the function over the parallelepiped with the vertices of the base at
and
and the vertices of the upper face at
, , and .
Figure 33
Solution
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
Task!
Find the Jacobian for the following transformation:
, ,
Evaluating the partial derivatives,
, , ,
, , ,
, ,
so the Jacobian is
where expansion of the determinant has taken place down the first column.