ENGR 2422 Engineering Mathematics 2

An Example using Jacobians in Multiple Integration


A major application of the Jacobian (Section 2.4 of the course notes) is to multiple integration over areas or volumes (Chapter 6).


Example:

A solid sphere of radius 2 m and centre at the origin has a non-uniform density given (in SI units) by

,  where  r is the distance from the point (x, y, z) to the origin.

Find the mass of the sphere.


 

For uniform density, the mass  m  of an object of volume  V  is just  m = rV.

For non-uniform density, the mass is

Cartesian coordinates are appropriate if the volume is a cuboid, in which case

dV  =  dx dy dz.

But for a sphere the limits of integration become very awkward unless the integral is transformed into spherical polar coordinates, for which

dV  = r2 sin q  dr dq df,  (where r2 sin q  is the Jacobian of the transformation from Cartesian to spherical polar coordinates).

 

In this example,

Therefore the mass of the sphere is

m  =  80 kg.

Note that the mass is finite even though the density increases without bound as one approaches the origin!


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Created 2001 02 01  and last modified 2004 04 02  by Dr. G.H. George