Tensors for Beginners 4: What are Covectors?

These are really tedious to make... I'm starting to lose steam. I'll make sure I finish this series, but I'm not sure how much I'll be able to manage afterward.

[ismaeel]

I'm a PhD student, honestly your videos are a godsend for me, your explanations are so good and the number of views also testifies to that fact. Thank you very much, looking forward to watching the rest of the series.

[MrCritical]

I am genuinely crying right now. As an engineering student minoring in physics, I am struggling so much with my general relativity classes. These videos explain everything clearly and simply. Your ability to break down concepts into simpler form reminds me alot of Feynman. Keep it up man +1 sub

[miguelaphan]

...¡ jesus! GOOD must bless you, for the very first time in history, there was someone capable of graphically explain what a covector is in the context of tensor analysis...thank you so much....please, keep doing this for differential forms and odd creatures like that...

[cherma]

Man I know these are really tedious derivations but for the sake of completeness and accurate understanding of what is going on I need this. So thank you chris.

[philandthai]

This is really jaw-dropping. I had no idea! Why didn't anybody tell me! How can I get to be 67 years old and still not know this, LOL. Excellent video. I'm watching the whole series in one big swallow, gulp.

[kshitijjha]

You are great. I am soo much interested in General Relativity but didn't know Tensors. I was waiting for almost 2 years to find something like this.

[BlueSoulTiger]

Congrats! eigenchris. Your vids are models of clarity and quality exposition. You have done the universe a big favour ;) Thank you.

[signorellil]

In the end we're all very happy you've stuck to doing this. Thanks from all your fans!

[cat-.-]

You achieve StatQuest level clarity in explaining algebra! I cannot think of a good enough complement for you! I owe you a lot for these videos!!!!

[carlosantoniogaleanorios]

Awesome videos. I just wanted to comment that colourblind people (like me) struggle to see the red as different from black. Red colour blindness is by far the most common sort of colour blindness, so if you could try to avoid the red-black pair when looking for contrast there will be about 1% of men that will be very thankful.

Just trying to give a tip that would make your videos even better than they already are. Thanks for the amazing work.

[MultiAblee]

like my dude that graphical explaination of covectors is a thing of beauty thank you for putting in this work

[amritanshraghav]

Amazing video series! I have tried to teach myself Linear Algebra and Tensors using a lot of online resources and yours has been the best! As an example, I had of course read about Linear Functionals and map from V -> R but the way you presented it here as a row vector being a function that takes a column vector was a complete 'Aha' moment for me. Thank you so much!

[ajaypotdar]

Thank you so much for these videos!!! I can't even begin to imagine the amount of effort it must have gone into making it so comprehensive and at the same time illustrative. You're amazing 👏👏

[karankhm]

Thanks, Chris. These videos are helping me a lot. I am a Master's student but was really taught bad in these tensors and your videos are helping me to understand clearly.

[peterrobinherbert]

I have no doubt that these videos are tedious to make, but they are absolutely brilliant. I have never seen this explained so well.

[guancongm]

these are some of the best explanations i have seen so far! thank you!

[jordansmirnov]

😯I studied these concepts on many books, I spent hours and hours to try to understand them without any result... It's been sufficient to watch your videos for a few minutes and Now, not only I understand them, but I can see them! Thanks a lot!

[AnanyaChadha]

I can't imagine how long these videos must take, the effort & hard work is so clear. THANK seriously, I'm taking a class that's teaching us about tensors and I didn't understand at all and was considering dropping it in school. But the only thing getting me through it is your videos so I'm sticking with it! So much better than any textbook I could find. Thank you so much
p.s. how did u learn all this stuff? is there a textbook you'd recommend? just curious bc your understanding is so stellar

[connorbeaton]

I needed this and didn’t know I needed it. Tasty vid.

[zestyorangez]

This really helps explain why a vector dot product with itself is related to its magnitude as well, cool.