Mathematical Methods of Physics

Mathematical methods have come a long way in making physics more interesting, simple and easier for us to understand. The different areas of physics had many challenges with respect to mathematical methods which scientists have been able to overcome with great success. Newton gave us a set of laws that were very easy to use as they applied just two magnitudes at every instance (like mass and force). He took the world by storm when it came out that these rules always applied in the real universe. “Mathematical Methods of Physics” is a science in which the knowledge and use of mathematics are applied in physics. Mathematical methods are used to solve problems in other fields too, but they are often more central to physics than, say, chemistry or engineering. It has been said that “physics is the language of nature” because all matter and phenomena are governed by physical laws. The mathematical descriptions through which we understand these physical laws often affect how we visualise our world and react to it, giving it a quantitative basis rather than just verbal descriptions.  In general terms, mathematics as a whole connects with every aspect of everyday life from architecture to cooking.

Overview on Mathematical Methods of Physics:

There are different types of mathematical methods used by physicists:

Mathematical methods of physics deal with the mathematical models that make up the structure or “language” of physics. These mathematical models provide a framework for analysing problems in physics and represent the central ideas that physicists use to analyse and explain physical phenomena, from elementary particles to the Universe itself. There are three levels of mathematical methods:

Basic Mathematical Methods:

Most physics textbooks begin their coverage with a series of simple equations. These equations are very much like those found in algebra or geometry, but they have been abstracted for use within the context of physics or science.  These basic methods include: circles, triangles, spheres and cones, etc.  These forms make it possible to create complex shapes and objects in space.  Third dimension geometry is based on the concept of vectors, which represent the motion of points in space.  By using these methods, it is possible to determine the relative positions and orientations of objects at a particular point in time.  This is called classical mechanics.  

Higher level Mathematical Methods:

There are various methods used by physicists to solve higher level problems which cannot be solved through basic methods. These problem solving techniques include:

Complex Analysis : This method solves problems by examining infinite or otherwise very large or complicated expressions with many terms. Mathematical methods of physics lecture notes will help you in understanding the basics of physics. The method of complex variables can be used to solve problems in advanced physics which cannot be done by the use of any other mathematical method. In order to understand complex analysis, one must first be familiar with complex numbers.  A complex number is an ordered pair (x, y) consisting of a real part x and an imaginary part y such that x2 + y2 = 0.  

Vector Analysis : It is basically a study of the transformation of physical quantities under change in space or time. Vector analysis is used to determine the relative position and change in displacement between two points across time (space-time). 

The mathematical methods of physics deal with the vectors in the space-time continuum. Different vectors can be combined or added together to form a vector grand or vector product, which describes a quantity that does not change when it is multiplied by another quantity.  A vector is a quantity in three-dimensional space.  This implies that all physical quantities can be expressed through vectors even though they exist in only two dimensions (space). In order to determine how two physical quantities related to each other change over time, physicists use vector analysis. The concept of zero and infinity plays an important role in understanding how magnitude affects changes in different physical quantities. 

Vector diagrams: The geometric interpretation of mathematics used to aid visualisation and problem solving is called vector diagrams. Mathematical methods for physics and engineering use vector diagrams to show the magnitude and direction of force, velocity, acceleration, etc. Vector diagrams are used to clearly show the relationship between quantities and the direction in which they are changing. Mathematical methods in the physical sciences are used to help visualise the equations that describe and explain physical phenomena. Vector diagrams can be used to explain and prove relationships between vectors, quantities, and their directions of change. 

Waves : This method involves treating physical quantities as forms of energy. For example, a sonic wave is a longitudinal wave in which energy is transferred as vibration or alternation of mechanical motion.

Conclusion:

All the above-mentioned topics are covered in higher level physics to solve complex problems. Mathematical methods of physics deal with the mathematical models that make up the structure or “language” of physics.The questions asked in competitive exams are not very tough as we have ample time to prepare for them. All we need to do is read the subject from the best books available and include as many questions from this topic as possible.