Forget teacher, even scientist, Prof. Balakrishnan is a "Sage", as his personality, knowledge, teaching style , illuminating power to every corners of truth reveals him. He knows the "Truth", it looks like that these are not formal classes but "The process of unfolding of truths, the pure theoritical knowledge". Call him "Reeshi Balakrishnan", and time will make people understand value of these lectures .
This is a delightful introduction to 4-vectors. The physics is not lost due to the mathematics as is quite often done in lectures on this topic. Beautifully done!
Speaking of "contravariant vector" or "covariant vector" is slightly an oxymore! Vectors and Covectors are INVARIANTS "geometrical" objects, intrinsic to any particular basis to decompose them! More precisely a Covector is a Linear Form that eats vectors and gives a number. What should be said correctly then when we chose a basis of vectors of the vector space, is that an INVARIANT GEOMETRIC OBJECT, as a VECTOR for instance, has CONTRAVARIANT COMPONENTS with regards to the Group transformation considered. But also has COVARIANT COMPONENTS! In 2D usual geometry they correspond to parallel and orthogonal projections! And the same for a COVECTOR that has COVARIANT COMPONENTS, but also CONTRAVARIANT ONES. That's all, but it is DIFFERENT than a tricky oxymore! Nevertheless further abuse of notation can be justified by the fact that the ith dual basis covector of the vector basis, in fact extracts the ith component of a vector when applied on it.
The (ds)2 given of the sphere is NOT necessarely the one of an arc portion of a great circle. Its a general one. Otherwise there would precisely be no dependantce of g22= sin(theta) in theta !
An important correction: Although the g's being constant throughout the space implies a flat space, the g's not being constant (i.e. depending on the coordinate values) does not imply curvature. E.g. the g's are not constant in a flat space coordinatized using polar coordinates. To detect curvature you need the curvature tensor or its equivalent, which essentially tells you whether or not your metric coefficient matrix can be mapped to the flat space one. Obviously this is not a GR lecture so Prof didn't go into detail on this, but it is an important point to emphasize because it's often misunderstood at first.
Forget teacher, even scientist, Prof. Balakrishnan is a "Sage", as his personality, knowledge, teaching style , illuminating power to every corners of truth reveals him.
He knows the "Truth", it looks like that these are not formal classes but "The process of unfolding of truths, the pure theoritical knowledge".
Call him "Reeshi Balakrishnan", and time will make people understand value of these lectures .
Check Leonard susskind's videos on SR too.
This man allows us to stand in awe of nature. He also allows us to appreciate the genius of man's unraveling of nature's secrets. Thank you Dr.
This is a delightful introduction to 4-vectors. The physics is not lost due to the mathematics as is quite often done in lectures on this topic. Beautifully done!
I totally agree
Rock solid concept building professor 😀
Speaking of "contravariant vector" or "covariant vector" is slightly an oxymore! Vectors and Covectors are INVARIANTS "geometrical" objects, intrinsic to any particular basis to decompose them! More precisely a Covector is a Linear Form that eats vectors and gives a number. What should be said correctly then when we chose a basis of vectors of the vector space, is that an INVARIANT GEOMETRIC OBJECT, as a VECTOR for instance, has CONTRAVARIANT COMPONENTS with regards to the Group transformation considered. But also has COVARIANT COMPONENTS! In 2D usual geometry they correspond to parallel and orthogonal projections! And the same for a COVECTOR that has COVARIANT COMPONENTS, but also CONTRAVARIANT ONES.
That's all, but it is DIFFERENT than a tricky oxymore!
Nevertheless further abuse of notation can be justified by the fact that the ith dual basis covector of the vector basis, in fact extracts the ith component of a vector when applied on it.
Great teacher
Thank You Sir
The (ds)2 given of the sphere is NOT necessarely the one of an arc portion of a great circle. Its a general one. Otherwise there would precisely be no dependantce of g22= sin(theta) in theta !
You can put ds value in Euler Lagrange eqn and you will see the magic
An important correction: Although the g's being constant throughout the space implies a flat space, the g's not being constant (i.e. depending on the coordinate values) does not imply curvature. E.g. the g's are not constant in a flat space coordinatized using polar coordinates. To detect curvature you need the curvature tensor or its equivalent, which essentially tells you whether or not your metric coefficient matrix can be mapped to the flat space one. Obviously this is not a GR lecture so Prof didn't go into detail on this, but it is an important point to emphasize because it's often misunderstood at first.
I would suggest lectures by R shankar (Yale University) for special theory of relativity then come back and have a look at this