In physics we deal with all the real-life problems and to express these problems we need a language, that is where mathematics comes into play. Mathematics play the role of medium between the real world and a piece of paper.
Mathematical physics is a branch of physics in which we study about the involvements of mathematics in physics and its role in physics, but not unlike other branches of physics namely mechanics, thermodynamics, optics etc. this branch has no independent significance, it kind of lives in the shadows of all the other fields but is also the most important tool of all the physics in the nature.
If talking qualitatively, there are topics in mathematics which are specially made for physics, e.g. calculus.
Topics of mathematics that are studied in physics.
Although every topic of mathematics has its application in physics, but saying just this doesn’t gonna help people so let’s start analyzing topic by topic.
- Linear algebra
Linear algebra is simply the combination of algebra and geometry. As both individually deals greatly with the problems of physics but discussing that will be a too basic thing to do. As we know problems in many fields of science involve the solution of the sets of linear equations either algebraically or by plotting on a Cartesian plane.
e.g. we found x=2 and y=-3 as a solution for a set of equations then we can think this solution as a point (2, -3) in the xy plane.
Some topics of linear algebra that are used for solving the set of equations are
Matrices and determinants: – this is itself a chapter and a very wide one of them. On just outlining the topic, matrices are the representation of data in an array format and have their vast applications is solving equations while on the other hand determinants can be designated as the magnitude of the matrices.
2.complex numbers
Complex numbers are the numbers that represents the combination of the real numbers and the imaginary numbers.
The use of this topic in physics has its own philosophical significance. This part of physics is not only useful to help in calculations but is also depicts that physics can have some quantities to represent whose index may not lie in the real world, there can be imaginary quantities to represent too.
Being a combination of real and imaginary quantities complex numbers are of great practical use, the complex numbers often simplifies setting up and solving vibration problems in either dynamical or electrical systems, and that is just one of many uses of complex numbers and notations.
3.series and sequences
The sequences are the arrangements of data which follow a certain criterion it (doesn’t matter that of what type the data is it just have to follow a common characteristic), when any mathematical operation is inserted between the terms of a sequence then it is known as a series.
The sequences a series have many of their applications in physics here is an example to prove it,
suppose there are charged particles placed at points x=0,21/2,41/2,61/2……. With negative charge and the positive charge particle is placed at x=1,31/2,51/2…… and we want to find the total force acting on the charge particle at x=0. Considering all the charged particles to be unit in magnitude then the total force at x=0 will be
F =1-(1/2) +(1/3) -(1/4) +(1/5) …….
By solving the series, we can find the total force acting on the charged particle at x=0.however solving the series will have its own concepts, but let’s not get to that now.
4.calculus
Being invented by Sir Isaac Newton this branch of mathematics is specially invented to be used in physics. There is no topic in physics which is free from this topic of mathematics. Mechanics, Quantum physics, optics, thermodynamics, and what not. Every derivation which is based on a real-life process has involvement of calculus.
Differentiation and integration in vectors is bold example, where the origin of multiple integration takes place.
Maxwell’s popular equations would not be present if there was no calculus(integration).
The Laplace transform and the Fourier transforms are the applications of the differential equations which are the key concepts for the network theory.
On combining the calculus with the other topics of mathematics we have many other topics that plays important role in mathematical physics namely
- partial differential equations
- functions of a complex variable the calculus of variations
- series functions of differential equations
and many more.
5.probability and statistics
Whenever deal with data and its management we have to get through this topic named statistics and probability, it has nothing to do with the type of data, and physics is a subject of data whether it is physical or mathematical.
“Mathematics and physics are just two faces of a coin inseparable yet different and that’s is the most beautiful thing about this combo.”
~ harsh
the physicsVesre 