Tensors Explained Intuitively: Covariant, Contravariant, Rank

[EugeneKhutoryansky]

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[FredyeahEternal]

As a hobbyist mathematician you have no idea how valuable these videos are, please dont stop making them, you're helping people be smarter

[ianpool]

I've spent so much time trying to find a simple explanation of covariant and contravariant vectors online, and in the first 3.5 minutes you've managed to out perform anything I've come across. A well deserved round of applause to you, Eugene! Keep up the great vids!

[joshedwards]

This channel is honestly top notch. Most resources are either too simplified to the point where they are not useful to someone who actually needs to learn this material, or they are so dense that a new learner gets lost in the details and misses the big picture. You do a great job at making the point clear (with the aid of amazing visuals) while also keeping everything accurate. Seriously, this is world class educational material. Get more famous!

[EugeneKhutoryansky]

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[black_wolf]

The professors I had in the university while doing my Bachelors all failed to explain the concepts of covariant contravariant in an understandable manner. You have done what they have failed to do in less than 12 minutes! :D

#RESPECT

[EugeneKhutoryansky]

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[MrJesuswebes]

Just a humble piece of advice: I think music should be more "subtle". Orchestral music is beautiful but I think it can "bother" a little when you try to concentrate on explanations. Of course: this is my point of view, of course.

[MrTiti]

our great classical music adds so much drama to on otherwise sober topic

[AndrewBrownK]

FINALLY A HELPFUL VISUAL REPRESENTATION!! I’ve been stuck on intuiting covariant vectors for YEARS! I think I get it now, it’s the *components* of the vector that are really covariant or contravariant, not the invariant/intrinsic vector itself

[umeng]

Having a good instructor makes a night and day difference when learning more advanced subjects. Great video. Making the jump from just dealing with vectors to tensors trips up a good number of people.

[tiuk]

Your channel should be promoted by some other famous channels, like Vsauce. Your videos are just too good. 3Blue1Brown got promoted this way. Maybe one day, this channel will as well.

[kimweonill]

Your combination of graphics, content and music is otherworldly 😊

[probiner]

I was looking into tensors 3 days ago and couldn't wrap my head around them and your video nailed it for me! Thanks a lot! Let me see if you have one on Quaternions, your skills might just finally break the wall for me to grasp how they are beyond Axis/Angle rotation and why if the axis is not normalized with a quaternion I get a skewed transform!

Keep up!

[Steven]

I've literally spent several years trying to understand tensors through self-studying to no avail. Your videos are the most intuitive and easy-to-understand way I've found and for the first time, I actually feel like I have a good understanding of tensors.

[JayLikesLasers]

Excellent introduction to tensors. It's funny how you could complete a whole masters or PhD and never see these any more than a 2d drawing of these mathematical objects, but then a video comes along and in under 12 minutes shows you what it took so long to wrap your head around to imagine.

[MrRobertT]

Eugene, your videos are absolutely incredible. Thank you for doing such a great job making things so well-explained and intuitive!

[parroj]

Really good video, you've done that people can visualize something which many professors didn't get in many years with their students and tried to explain as a teachers a visual concept with lots of usefuless words and few quality visualizations. Thanks

[kbwsoikat]

You tube amazing because of the people like you who believe that knowledge should be free.

[gruminatorII]

Absolutely phenomenal video, i really wish we had these to study 8 years ago. I finally understood the difference between co and contravariant .... before i just knew the definition