Mathematics is central to all engineering disciplines, including aerospace engineering. After all, it underpins every computation that allows aerospace engineers to better understand physical situations and optimize aircraft designs. Therefore, it is worth dedicating a week to discuss math.
The first main topic of this discussion is calculus, which has two central concepts. The first of these concepts is differentiation. Geometrically-speaking, differentiating (or taking the derivative of) a function allows us to find an equation giving the slope of the tangent line to the graph of that function at every point. This slope is a rate of change. For example, if there is a function, y(t), that gives the position of an object, y, at any time t, then the derivative of y(t), denoted by dy/dt, gives the change in position per change in time, or velocity, of that object.
The second central concept of calculus is integration, which seeks to find the area under the graph of a function. To do this, we imagine that there are many rectangles that roughly cover the area under the graph. Then, we can notice that if we have infinitely many infinitely thin rectangles, then the total area of all of the rectangles is exactly equal to the area under the graph of the curvy function. When I first learned integral calculus, the idea of summing up the areas of infinitely many infinitely thin rectangles seemed impossible, almost magical. But sure enough, this is possible by using a mathematical tool called a “limit.” The concepts of integration and differentiation are related in that taking the integral of a function is the reverse of taking its derivative and vice versa. For example, the area under a velocity vs. time graph gives the change in an object’s position.
The second main topic of today’s discussion is differential equations, which builds on concepts from introductory calculus. Basically speaking, a differential equation is an equation that relates a function with one or more of its derivatives. For example, the equation dy/dt + y = 0 is a differential equation. In particular, this example equation is telling us that if we add the rate of change of y to y itself at any time t, then we get zero.
One key differential equation that I learned in my first-year aerospace engineering course was Euler’s equation, expressed as dp/du = -ρ*u, or more commonly, dp = – ρ*u*du. This equation explains how all fluids work (albeit in very simplified, ideal conditions)! Here, “dp” refers to a tiny change in the pressure of a fluid, “du” refers to a tiny change in that fluid’s velocity, “u” represents the velocity itself, and “ρ” denotes the density of that fluid. What this equation says is that if a fluid’s velocity increases moving from one point in space to another, then that fluid’s pressure decreases from that starting point to that ending point. Moreover, this decrease in pressure is weighted by the fluid’s density and instantaneous velocity.
Calculus and differential equations can be complicated subjects, and it can take many semesters of studying to master them. That said, I believe that in-depth studies of these subjects are worthwhile because they can be powerful tools that provide a language through which one can make quantitative discoveries.