In practice, extremes of scalar fields in n variables \(x_1 ,\dots , x_n\) are usually to be determined under constraints. Such constraints can often be described as zero sets of partially differentiable functions in the variables \(x_1 ,\dots , x_n\) . There are essentially two methods to determine the sought-after extreme points and extremes, the substitution method and the Lagrange’s multiplier rule. With the substitution method, nothing new happens, we have even already considered a first example in the last chapter; however, the substitution method is not as universally applicable as the multiplier rule of Lagrange, which, on the other hand, has the disadvantage that it often leads to only difficult to solve nonlinear systems of equations.