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The concept of a system in engineering can be understood in a broad sense; we can analyze the simplest contraption as systems as long as we have an input, an output, and a medium for the input. We have examined ways to analyze systems while we were solving general engineering problems; however, we were sidestepping the act of characterizing the system itself with an equation or model—we only considered actions within the system. Although those pieces are important, we are missing the crucial mathematics in order to describe systems effectively. For instance, how do we mathematically model our now-familiar spring and mass? How about the swinging of a pendulum?
Perhaps those are too simplistic, but the idea remains the same regardless of complexity. For instance, the method in which we go about modeling the spring and mass would be carried out in a similar manner as if we were modeling a circuit. Two major tools we would need come from well-studied areas of mathematics: linear algebra and differential equations. There are just a few pages left, and so this is obviously beyond the scope of this book.
6.1 A SUMMARY AND WHERE TO GO FROM HERE
Throughout this text, we covered a large portion of the mathematical knowledge needed for engineering coursework, but we could not hit all of the important topics. We began by discussing what it meant to abstract physical situations and solve problems in a simplified space, then explored different problem-solving tools and fundamental results that simplified our calculations. After looking at the techniques to solve problems, we turned to the engineering perspective by collecting what we knew and applied them to the concept of “signals and systems.” For now, we need to cut the exploration short before talking about the more advanced half of the story.
In undergraduate studies, the differential equations class was described as the course that provided the underpinning theory to succeed later in the engineering curriculum. Differential equations are equations that involve a function and its derivatives, making them the logical next step after calculus and this text. In addition, learning about matrices will be useful when solving large-scale problems both within and outside the classroom.
A world of mathematics exists beyond this book, go out and explore!