From Theory to Reality: The Intersection of Physics and Mathematics
This article elaborates on some of the branches of math relevant to physics and explores their scientific applications.
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Physics calls to mind many simple demonstrations that most people have seen, from Newton’s cradle to electrified frizzy hair. But the subject is also associated with intricate and extreme natural phenomena: quantum tunneling, supernovae, and more. We all understand that science at that level requires difficult mathematics to establish a rigorous framework, but most high schoolers are yet to explore the mathematical subtlety of higher-level physics.
For physicists, certain branches of math are more relevant than others. The most obvious of these is calculus. For those who have yet to learn calculus, a brief summary of the field is that it studies how quantities change in relation to one another. Stuyvesant offers AP Calculus AB and BC, as well as higher-level courses like Multivariable Calculus and Complex Calculus. Students who have taken AP Physics C have first-hand experience with how effective calculus can be in illustrating scientific concepts. The course re-derives the principles of motion and electromagnetism using calculus, allowing for more in-depth study. For instance, applying calculus in classical mechanics can help to describe the position, velocity, and acceleration of an object at particular points in time. It also serves as the foundation for definitions of useful quantities like work—the action of force over a distance—as well as various equations for modeling rotational motion. Without calculus, one could only work with the most simple cases of these, which are unrepresentative of their true dynamic nature. The endless fluctuations and changes of the universe necessitate the use of calculus in physics.
Linear Algebra, another post-calculus math course taught at Stuyvesant, is a branch of mathematics that deals with linear equations, vector spaces, matrices, and more. The concept of vectors is crucial to physics, as they simplify operations on quantities with multiple components and dimensions. A vector can be defined simply as that which has both numerical magnitude and direction. High school physics classes usually study the application of vectors in classical mechanics and electromagnetism. For example, an electric charge A will produce a force on any other arbitrary electric charge B. The electric field of A is a function that describes how the force on B behaves based on the position of B. The electric field points from B toward A, and a vector is used to indicate this direction. Vectors allow us to extend concepts such as a simple two-dimensional model of the electric field into the real three-dimensional world.
Likewise, group theory is the study of algebraic structures that share similar properties. A group is a set of elements in which there exists a specific operation between any two elements of the set, and it must satisfy certain conditions. The operation must be associative, an identity element must exist, and every element must have an inverse. A familiar group is that of real numbers under addition. Real number addition is associative and has an identity element of zero. Every real number has an additive inverse, a number to which it can be added to produce zero. Group theory is incredibly important to higher-level physics because it formally establishes the concept of symmetry beyond a simple geometric approach. Symmetry occurs throughout the universe—in atomic nuclei, the Standard Model of particle physics, and the physical structure of crystals. Describing the remarkable properties of these phenomena requires the special language that group theory provides.
For those interested in physics, it is important to become acquainted with the utility of mathematics. The physicist Eugene Wigner was fascinated by the “unreasonable effectiveness of mathematics” in our study of the natural world. This effectiveness exists because mathematics has been forged from a thorough understanding of logical principles, resulting in a tool that all physicists must have handy. Calculus, linear algebra, and group theory, among other fields of math, make it possible to accurately predict the behavior of physical systems. Students of both mathematics and physics must realize the power that they hold to uncover the secrets of our universe. Similarly, educators must try to convey this in classrooms, for the students of today are the great intellectuals of tomorrow.