Video 10 - Covariant Basis Examples

[@MikeMagTech]

I would like express my gratitude for this superb course. Your excellent presentation, care, and attention to detail make the subject matter surprisingly approachable.

[@imrans7545]

Loved it. Your teaching skills are perfect. And you are not wasting student time in drawing diagrams on the board, so you have managed to condense what can be taught in many many hours in minutes. Thanks a lot !

[@tensorcalculus822]

Thanks for your comment. I work hard not to waste video time. It's nice to have it recognized.

[@RAKESHMANDAL-s4c]

This course is a masterpiece. Thanks

[@Tester-w2n]

Thank you. Very clear explanations

[@brianyeh2695]

Thank you very much for this video. I have never seen someone explaining by moving and scaling the coordnates interactively! This way helps bring better intuition!

[@eastofthegreenline3324]

Really authoritative and helpful series. Thanks.

[@thevegg3275]

You said that 19 minutes and six seconds that these are covariant basis vectors, but I look at them, and all I see are basis vectors . what makes them any different than contravariant basis vectors and how can you prove that by taking me back to the graphical illustration of dual bases vectors, which describes covariant components?

[@tensorcalculus822]

Video 17 includes a visual demo that compares covariant and contravariant basis vectors for affine and plane polar coordinates. You should be able to see how this extends to the other coordinate systems. What you will find is that for orthogonal coordinates, where all the basis vectors are mutually orthogonal, each contravariant basis vector points in the same direction as its covariant basis counterpoint. Each pair points in the same direction, but their magnitudes are reciprocals of each other. If they are unit vectors then they will be the same. It is only when the system is skewed that they point in different directions. Hope this helps.

[@joeboxter3635]

​​​​@@tensorcalculus822For almost a year, I have been trying to understand where the r in the theta comes from and why if that being the case is there no theta (by symmetry) in the r direction, if you paramertize by arc length. You explained and corrected my misunderstanding in 10 min with your beautifully done presentation. I am *so* grateful! I also want to say, it's a crime that calculus 3 is not taught using tensor calculus. I memorized so much and trusting memory is very poor substitute for actually grasping the rational. Kudos to you for this as well because my plan is to finish this series and go back to Calc 3 and learn using tensor analysis.

[@Tester-w2n]

Thank you. Very clear explanations