Hard water
Water's hardness is determined by the concentration of multivalent cations in the water. Multivalent cations are positively charged metal complexes with a charge greater than 1+. Usually, the cations have the charge of 2+. Common cations found in hard water include Ca and Mg. These ions enter a water supply by leaching from minerals within an aquifer. Common calcium-containing minerals are calcite and gypsum. A common magnesium mineral is dolomite (which also contains calcium). Rainwater and distilled water are soft, because they contain few ions.
Stiffness matrix
By defining the vector "F" with components "F" = formula_5, the coefficients "u" are determined by the linear system "AU" = "F". The stiffness matrix is symmetric, i.e. "A" = "A", so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that the system "AU" = "F" always has a unique solution. (For other problems, these nice properties will be lost.)