From [[Dimensional Analysis]]. Our task is to find a relation $u$ in order to determine some variables $d$ (dependent variables) from other independent variables denoted $i= (i_{1},\dots,i_{M})$: $d=u(i_{1},\dots,i_{M}).$ Whether we can find this function $u$ depends on the mathematical statements that connect the variables. ### Recipe 1. Express the dimensions of all variables $d, i_1, . . . , i_M$ in terms of the fundamental dimensions. 2. Write down a general product combination of the independent variables for the dimension of the dependent variable, $[d] = [i_1^{n_1} i_2^{n_2}...i_M^{n_M}] = [i_1]^{n_1}\ldots[i_M]^{n_M}$From this, derive an equation for each fundamental dimension that appears. 3. Solve emerging system of linear equations. We can the write $[d]=[d_{s}, \pi_{1}, \pi_{2},\dots,\pi_{N}]$where $d_{s}$ and $\pi_{i}$ are independent products of powers of the independent variables such that $d_{s}$ has the same dimension as $d$ (**we say it is a scale for $d$**) and the $π_{l}$ are nondimensional. 4. The reduced, nondimensional problem then reads $d=d_{s}(\tilde{u}(\pi_{1},\pi_{2},\dots,\pi_{N})).$ See [[Scaling Procedure for Scalar Ordinary Differential Equations]].