Fourier Analysis
When sinusoidal waves are added together, the resultant wave is the sum of the constituent waves' amplitudes. This is called a superposition. Fourier transform allows one to separate a superposition into the trigonometric functions that make it up. An amalgamation of seismographic data, for example, can be decomposed into its constituent frequencies (e.g. if a seismograph collected information about earthquake body and surface waves that overlapped).
Vector Fields
Vector fields, which allow for the representation of a collection of vectors in space, have applications in multivariable calculus. They can be utilized to represent the strength of gravitational, electrical, and magnetic forces at various locations in space. Vector fields can also model the flow of heat and fluids. The below diagram shows the vector field of the gravity exhibited around a massive object.
Hyperbolic Functions
Hyperbolic functions allow for the mathematical expression of hyperbolas and are used to describe the geometry of the hyperbolic plane. Complex numbers form the relationship between the trigonometric functions (sine, cosine, tangent) with the hyperbolic functions (sinh, cosh, tanh). The equations for the latter are derived with Taylor series. These appear in physics, for example: in special relativity, hyperbolic functions create a relationship between an event's position and time coordinates.