We consider a typical heat conduction problem. At time t=0 we let an amount of heat e concentrated in a point in space diffuse outward in a region with original temperature zero.
If r denotes the radial distance from the source and t is the time, then the problem consists of determine the temperature u as a function of r and t. We easily realize that the temperature u depends on t, r and e. It is also reasonable that it depends on the heat capacity c of the region as well as the speed with which the heat diffuses outward (the heat diffusivity k). We therefore assume a physical law on the form
relating the fundamental quantities t, r, u, e, k, c. These quantities have the dimensions
Symbol: 
Quantity: 
Dimension: 
t 
time 
T 
r 
length 
L 
u 
temperature 

e 
energy 
E 
k 
heat diffusivity 
L^{2}/T 
c 
heat capacity 
The dimension matrix A then becomes
We thus have m=6 (number of physical quantities), n=4 (number of fundamental dimensions) and r=4 (rank of the dimension matrix). This implies that we can construct mr=62=2 dimensionless quantities. We therefore look for two linearly independent solutions of the system
If we let
we get
that is
If we choose v=1/2 and u=0, we get one solution
If we choose u=1 and v=3/2, we get another (linearly independent) solution
These two solutions give the two dimensionless variables
According to the Pitheorem we then conclude that there is an equivalent relation
that is
If we finally solve for u, we get