When we practice problem solving, the purpose is not merely to find the answer; rather, we solve a problem with the goal of learning how to solve it. Beware: Just because you have solved a problem doesn't necessarily mean you know how to solve it. For practice to be effective we must not only understand what we are doing, but why we are doing it. The twin goals of what and why are best met if we have a systematic procedure for attacking problems. This document presents and illustrates one such procedure.
The Fourier transform can be interpreted as an extension of the Fourier series from periodic to nonperiodic functions. This document introduces the mathematics and use of Fourier transformations. The presentation includes descriptions of simple transforms, convolution, and how the Fourier transform combines with sampling theory to yield Heisenberg's uncertainty principle.
In a traditional approach to balancing chemical reactions, we apply conservation of atoms to balance the number of reactants against the number of products. This requires us to know which reactions are occurring, but in many practical situations we do not know all the reactions. Nevertheless, if we can simply list all reactant and product species, then we can still find a complete independent set of balanced reactions. This can be done by using the list of reactants and products to build a formula matrix, and then performing a singular-value decomposition (svd) on that matrix. This paper describes and illustrates the procedure.