Lagrange multiplier's with open source software

Abstract

Many students have a hard time with Multivariable Calculus because of its abstract theoretical foundations and, at the same time, geometrical skills involved. In recent years things have become even worse: new study plans have been designed for Engineering careers in which less time is available for mathematics. Some topics taught in second year calculus in old plans are now exposed in the first semester. In this way the student have very little time to mature the new knowledge.In the other hand in many textbooks advanced mathematical concepts are exposed in a too abstract way that does not help students to understand the topic.Fortunately many mathematical topics have been rethought and exposed in new ways more suitable for students needs. Moreover now we can use for free computer software to make calculations and to draw geometrical representations unimaginable not long ago.We present here a geometric intuitive approach to the constrained optimization problem in a multivariable calculus course for engineering students. Open source software like Maxima and Asymptote is used for this purpose. The basic ideas of this presentation follow.- Rouche-Frobenius theorem shows how an undetermined linear equations system is equivalent to a parameterized linear variety.- Implicit Function theorem shows how an undetermined nonlinear equations system is equivalent to a set of parameterized non linear varieties.- Constrained optimization problem is equivalent to an unconstrained one if a parameterization of the constraint is available.- If such a parameterization is not available, the Implicit Function theorem provides a necessary condition known as Lagrange's Multipliers Method.- All the situations are very geometrical in nature and plotting software like Maxima and Asymptote let us to see what is around in the low dimensional cases.