I hope all of my readers are doing well as midterm season approaches. In this week’s blog post, I will try to cover a lot of information in a short amount of space. First, I will be building up intuition for a math formula that has staggering importance. This formula is called the Generalized Stoke’s Theorem, and very well may be my favorite piece of math. Then, I will explore how special cases of the Generalized Stoke’s Theorem can be used in physics and aerospace engineering.
We will start with the Fundamental Theorem of Calculus, which is usually covered in an introductory calculus class. This formula can appear in textbooks in several forms, but it all boils down to this equation:
∫[a,b] (df/dx) dx = f(b) – f(a).
We are saying that the integral of the derivative of a function over an interval [a,b] is equal to the function itself evaluated at the ending point of the interval minus the function itself evaluated at the starting point of the interval. Since the interval [a,b] is one-dimensional, the boundary of the interval consists of the zero-dimensional points x = b and x = a. Therefore, we can view the term “f(b) – f(a)” as a sort of integral of f(x) over the boundary of the interval [a,b]. This idea of evaluating a function at the boundary of an interval is important for what comes next.
Near the end of a course in multivariable calculus, professors introduce two theorems that look similar to the fundamental theorem of calculus, but also look a bit different because they have all kinds of symbols that aren’t present in Calc I. Indeed, these theorems deal with vector fields, which are basically distributions of many many vectors throughout two-dimensional or three-dimensional space. One of these theorems is called Stoke’s Theorem, which appears as this equation:
∫∂S F • dr = ∫∫S curl(F) • dS
On the left side of the equation, we have a line integral of a three-dimensional vector field F(x,y,z) over a closed curve (denoted by ∂S) that is the boundary of a surface S. In physical terms, this line integral is calculating the work done by a force field F on a particle that crawls along the closed curve ∂S. On the right side of the equation, we have a double integral that is calculating the flux of another vector field, curl(F), through the surface S. In physical terms, this flux integral is quantifying how much the vector field curl(F) flows through the surface S. The important fact about the right-hand side of the equation is that curl(F) is a type of derivative of the vector field F. Therefore, despite the heavy notation, Stoke’s Theorem is once again saying that the integral of a function over the boundary of a region is equal to the integral of the derivative of the function over the entire region.
The second of the important theorems in multivariable calculus is called Gauss’s Theorem or the Divergence Theorem, and appears as follows:
∫∫∂G F • dS = ∫∫∫G div(F) dV.
This equation is telling us that the flux of a vector field F(x,y,z) through a closed surface (denoted by ∂G) that forms the boundary of a solid region D is equal to the integral of a scalar function, div(F) over the entire solid region. The most important fact to explain about this equation is that div(F), also called the divergence of F, is another kind of derivative for multivariable vector fields. Therefore, the Divergence Theorem is also saying that the integral of a function over the boundary of a region is equal to the integral of the derivative of a function over the entire region.
In the 19th and early 20th centuries, the mathematicians Henri Poincaré, Édouard Goussat, Vito Volterra, and Éllie Cartan helped form a generalization of all of these important theorems from single and multivariable calculus. This generalization, indeed, is called the Generalized Stoke’s Theorem or Stoke’s Theorem on Manifolds, and appears as follows:
∫∂M ω = ∫M dω.
While this equation has a lot of complexity baked into it (including the concepts of manifolds and differential forms), it is essentially conveying the central theme that derivatives and boundaries are opposites of each other. Therefore, three semesters of calculus can almost be boiled down to this single equation.
When it comes to physics and engineering, the Generalized Stoke’s Theorem has many applications, most of which amount to boiling the generalized theorem down to one of its special cases. For example, if we wish to find the change in position, x(t), of a particle that has a known velocity function v(t) we can do the following integral:
∫[a,b] v(t) dt = x(b) – x(a).
Since, an object’s velocity is the derivative of its position, we are applying the Fundamental Theorem of Calculus (one special case of the Generalized Stoke’s Theorem) to say that the integral of velocity over time is equal to the change in position from the start of a time interval to the end of a time interval.
Meanwhile, much of aerospace engineering involves studying fluid flows. In the analysis of fluids, it can be useful to determine the mass of air that flows through a closed surface per unit time. To do this, we can apply the Generalized Stoke’s Theorem (in the special form of the Divergence Theorem) to say:
dm/dt = ∫∫∂G ρv • dS = ∫∫∫G div(ρv) dV.
In this equation, ρ is the density of air, v is the velocity of air, and ∂G is the closed surface that the air is flowing through. G is the solid enclosed by the surface ∂G. In computer modeling, it is often faster to calculate the triple integral than the flux double integral, making the Divergence Theorem (and by extension the Generalized Stoke’s Theorem) very useful.