Tensor Analysis for Physicist

Tensor Analysis is one of the most useful subject in higher mathematics. It seems to be very abstract to physics students because it is compact in notation. Every student always try to find traditionally from where tensor analysis begins. It is very useful due to its great generality, computational power and easy notation. Today it is very useful in Data Science and Machine learning also.

Tensors are some mathematical quantities which transform according to certain rule known as Tensor algebra rule. We will discuss some of rule here and guide the learner to know more deep about the subject. Tensor manipulations of indices are very simple and useful for physics students.

All physical quantities are coordinate independent so we require to model some mathematical framework for that. It is tensor analysis.

In the tensor analysis, we search always coordinate independent quantities for application in physics. We choose those quantities that have component transformation properties and quantities are independent of observer’s coordinate system. So tensors are ultimately defined as in terms of its transformation properties.

For simplicity let us begin with number systems, Numbers are the heart of all mathematicians. Numbers are abstract but represented by words one, two,…etc. We can relate numbers directly to real world. There are so many categories of numbers, Natural number, whole number, Integers, Real numbers and Complex numbers, Transcendental numbers like e and pi.

Numbers we can use infinite ways, example we can count population of a city or fruits in a basket. More concrete you can say 7 apples and 3 bananas in basket. But everything is not very simple suppose, I am asking distance of your college from your home and you are saying 20. But 20 what? So here is more information is required and you have to add another quantity also known as unit. 20 km makes a sense. These kinds of numbers some times known as scalars. Mass, volume and temperature are considered as scalars.

Suppose your college is very near to your house just 19/20 km far you required rational number to denote this. If someone asking is 19/20 km where so you can say in the east or west. You need to specify the direction also. Such quantities are known as vectors.

Suppose we are in a Krishna temple and we stopped to examine a beam in the side of temple corridor. I might ask what is the net load on this beam any one can give the answer so many newton in downward. Answer is very appropriate but now if someone is asking what is the stress in the beam?(I hope everyone is familiar with stress)

I will say. There are so many stress, three tensile and six shear stress. Which do you want to know? and which part of the beam you are interested? Now subject Tensor is become introduce because even vector can not give the sufficient answer of this question.

To explain this we will add new terminology called directionality.

To learn more about Tensor Analysis please watch my videos on youtube channel: https://www.youtube.com/c/aggrawalsirphysics

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