Contravariant, covariant and physical components of tensors

This video looks at the ideas of contravariant, covariant and physical components and how to convert the first two component types into physical component values. This is a necessary skill when working with tensors in non-cartesian coordinate systems because the physical component values are those that would be measured in a Laboratory.

[@nikhiliyengar9899]

Your video is incredibly clear and has EXACTLY the kind of explanations that I was looking for to understand physical components. Awesome job!

[@TensorCalculusRobertDavie]

Nikhil Iyengar Thank you Nikhil.

[@g3452sgp]

Very Straightforward and clear explanation about coordinate components and physical components . I can say this video is one of the most excellent one in this field ! Thanks a lot.

[@TensorCalculusRobertDavie]

Thank you.

[@Mikey-mike]

Excellent I love this lecture.

[@TensorCalculusRobertDavie]

Thank you Mikey.

[@georgeveropoulos2489]

Very. Nice. Work.

[@TensorCalculusRobertDavie]

Thank you.

[@georgeveropoulos2489]

do you have any ppt file with this ?

[@nathsujitkumar]

Very helpful video. Could you please suggest any reference for this? A paper by Daniel Frederick (PHYSICAL INTERPRETATION OF PHYSICAL COMPONENTS OF STRESS AND STRAIN) and another by C. Truesdell (The physical components of vectors and tensors) seem to give a little different expressions for physical components of second order tensors than what you derive at 11:14 in the video. However, both your derivation and the papers' derivation match for the orthogonal coordinates. But I wanted know the correct expressions in the case of general coordinates. Thanks!

[@TensorCalculusRobertDavie]

Thank you for your thoughtful question and for watching the video! I appreciate your interest in understanding the physical components of second-order tensors in different coordinate systems. The expressions for physical components of second-order tensors can indeed vary depending on the coordinate system used. In the case of orthogonal coordinates, the derivations tend to align more closely because the basis vectors are mutually perpendicular, simplifying the mathematical expressions. For general (non-orthogonal) coordinates, the expressions can be more complex due to the interaction between the coordinate system's basis vectors. The key difference arises from the metric tensor, which accounts for the non-orthogonality of the coordinate system. Here are a few references that might help clarify these concepts further: Daniel Frederick's paper: "PHYSICAL INTERPRETATION OF PHYSICAL COMPONENTS OF STRESS AND STRAIN". This paper explores the physical components in various coordinate systems, including non-orthogonal ones. The differences you noticed likely stem from how the metric tensor is incorporated. C. Truesdell's paper: "The physical components of vectors and tensors". Truesdell's work is foundational in the field of continuum mechanics and provides a comprehensive treatment of tensor components in different coordinate systems. For the expressions in general coordinates, the correct approach involves using the covariant and contravariant components of tensors, along with the metric tensor. The metric tensor 𝑔_𝑖𝑗 relates the physical components of the tensor in general coordinates to those in orthogonal coordinates. In summary, while the derivations in my video match those in the papers for orthogonal coordinates, in general coordinates, it is essential to account for the metric tensor and the potential non-orthogonality of the basis vectors. I hope this clarifies the differences you observed. For a more detailed mathematical treatment, I recommend reviewing the aforementioned papers and the sections on tensor calculus in general coordinates. Feel free to ask any more questions or request further clarifications. Thanks again for engaging with the content!

[@nathsujitkumar]

@@TensorCalculusRobertDavie Thank you so much for your clear and elaborate explanation. I will have a deeper look into this and try to clear my ideas using the references. Thanks for making such nice informative videos! Very much appreciated. Keep on doing this. I feel bad for myself that I discovered your channel so late. However, better late than never. I am now a subscriber and a regular visitor to your channel! 🙂

[@me_nalayak]

Can u suggest any good book for tensor calculas with better physical intuition???

[@TensorCalculusRobertDavie]

You could try, "Introduction to Tensor Calculus and Continuum Mechanics" by J.H. Heinbockel as well as the various videos on KZbin. Hope that helps?

[@TensorCalculusRobertDavie]

A lot of it comes down to working things out for yourself. That is why I hope my channel will help people trying to understand the concepts of this field.

[@me_nalayak]

Yeah thanks!!

[@TensorCalculusRobertDavie]

You're welcome.

[@nikhiliyengar9899]

A Brief on Tensor Analysis by Simmonds is a great intro with a heavy emphasis on components and physical meaning of tensors

[@alexrusnak7574]

You need a better mic)

[@d3faulta]

I agree

[@TensorCalculusRobertDavie]

I have since obtained a better laptop.

[@DaytonaStation]

good but sound is awful

[@TensorCalculusRobertDavie]

Thanks for that. More recent videos have been made with better sound.

[@DaytonaStation]

i am concerned about the sound. i replay my videos and the sound is perfect. I am wondering what is the problem.