Tensor Calculus For Physics Majors #1| Preliminary Vector Stuff part 1

This video is the first part of a series on tensor calculus based off of the book "Tensor Calculus For Physics" by Dwight Neuenschwander. I discuss vector operations using the kronecker delta, levi civita symbol, and the metric. We also go through a physics problem involving an inclined plane to demonstrate rotating a coordinate system.
Link to Tensor Calculus for Physics Book:
www.amazon.com/gp/product/142...
Check out this link for a better explanation on the x',y' coordinate method of solving the inclined plane:
• Inclined Plane Problem...

[@missingno9]

Your handwriting's really nice :D It's spelled "demands", not "demends". I think you could've unpacked the Levi-Civita notation for the cross-product explicitly so it would be clearer for other people. At around 39:10, Φ should not be 90 + θ but rather 90 - θ, since Φ decreases when θ increases. It works out though, because cos(90 - θ) = sin(θ), which you mistakenly put as cos(90 + θ) = sin(θ). With the second coordinate system, the way you handled Φ geometrically had some flaws as well, but the components turned out to have the right terms. But I'm just splitting hairs here. Great job on this video, and I'm looking forward to the next ones!

[@AndrewDotsonvideos]

Thank you for pointing this out. I was very hand waivey with that part.

[@heavennoes]

@@AndrewDotsonvideos I just realised that cos(90+x) = -sin(x), so shouldn't F_x' = |mg|sin(x)+|f| since the the -|mg| would cancel out the negative with -sin(x)

[@sour5blue]

K0’m V

[@devinlocke3691]

Andrew: "What's going on smart people?" me: *looks around my room for other people*

[@rajaradi802]

What's going on idiot people Me: looks around the room for other people

[@NovaWarrior77]

@Asyam Abyan You're doing my job for me!

[@naturematters08]

@@rajaradi802 ahahahahaha

[@ty6339]

@Asyam Abyan where do I find your motivation speechs

[@swagat14gameplays]

@Asyam Abyan dayum🤣❤️

[@Johncornwell103]

Originally subscribed for the physics memes, staying for the lectures.

[@ahmedd798]

Ok

[@selvakumar3k]

@@ahmedd798 xvS sz ZZ z's ZZ xx z

[@bingusiswatching6335]

Samee, this was so useful

[@BillPark-ey6ih]

same

[@jeanlucas2834]

same

[@ianprado1488]

My topology teacher's favorite joke was saying that general relatively is invariant under notation

[@AndrewDotsonvideos]

Ian Prado hahaha I’m going to have to remember that one

[@zes7215]

no such thing as jokx or not

[@AlchemistOfNirnroot]

???

[@KFlorent13]

I wish I could get it

[@unrelentingawesomeness7501]

@@KFlorent13 yea

[@AndrewDotsonvideos]

Okay so far the feedback suggests I need to speak louder when facing the whiteboard, and I was a little hand-waivey with the inclined plane problem (which I recognized while editing the video). I'll be sure to be more careful with my arguments geometrically in the future. I'll also include a reference in the beginning of each video to explicitly state what will be covered so you don't have to watch the whole video to find out. Was debating fitting the whole section on vectors into this one video. Glad I didn't!

[@GLPentAxel]

Andrew Dotson I'm almost a bit surprised how good this actually is! One more small thing you could consider to make it flawless is to put the audio through some "filtration" program to get rid of the static background noise. Looking forward to watching the rest, especially now that I just finished my exams!

[@sciencestararvinsinghk]

When's the next vid coming?

[@amensutenhotep1]

dude u should get more buffed! your vector views will increase 3 fold Lol

[@angelmendez-rivera351]

BLAIR M Schirmer Well, you are an arrogant one, for sure. You apparently do not understand the concept of "context." This video is not intended as an introductory course nor for beginners, nor is it designed for people who have no idea of what a tensor. He has other videos designed for that type of audience. This is not his incompetence, this your damn incompetence for not knowing how to search videos on a channel. It's incredible how people in 2018 were still not aware of how to competently use KZbin.

[@angelmendez-rivera351]

BLAIR M Schirmer Also, he DID mention the definition of a tensor very early on in a video, and then talked about it would be absolutely unhelpful to start talking about it immediately. So, looks like you have a bad case of not paying attention, or of being a dishonest prick.

[@trigocuantico]

0:00 Important Introduction ___________________________________________ The elements 4:53 Products of vectors and their intuition 6:29 Derivatives in vectors 7:35 the Gradient, Curl and Divergence ___________________________________________ What we do with them 9:20 we begin with coordinates 12:40 representation of a vector r in this coordinate idea 14:24 Basis 18:30 Dot product for the introduction of new concepts: Metric, Kronecker Delta ___________________________________________ Applying this products and ideas 20:28 Dot product between two general vectors 24:21 Cross product between two general vectors ___________________________________________ Quick note if you're not careful with angles 29:36 Comparison Dot product for x component and y component ___________________________________________ Physics problem 33:16 Block on plane intro 35:24 2nd Newton Law for the problem and the first way (non block-local x & y) 42:38 2nd way, we rotate our view so we are inclined with the plane (block-local)

[@teslathejolteon8007]

I think this is my favorite youtube channel already, so clearly explained!

[@ryangowen2816]

As a hobby student who came here through your meme reviews, this is absolutely the most helpful tutorial for tensors on KZbin. All the other tutorials assume you already know the foundations, but as a hobby student my foundations are full of holes, so it's nice to see things like the Kronecker delta defined by a logical progression than with a bunch of words that don't mean anything to somebody who doesn't have a good foundation to start from.

[@StNick119]

This was a refreshing take on how to interpret dot, cross products and the gradient. I've never been quite as good at applied maths as at pure, so I relish this video and the chance to improve my understanding. Looking forward to learning tensors, I've never seen them before!

[@hiunchen2222]

absolutely beautiful and concise lecture. Thank you very much! My hope to understand tensor calculus and GR brightens!

[@SirThorp]

This doubles as a really informative lecture and sleep aid. Thanks Andrew and I hope exam studying is going well!

[@kelvinlam5291]

Three years later, this is finally useful as I'm starting to self-study GR

[@michaelupdike-bz6rg]

Finally, I feel like I'm ready to learn this. So excited!

[@pifibbi]

For a moment there, I thought you said "Welcome to Tensor Calculus for Fitness". For a moment there, I was excited...

[@blurp8996]

I was honestly very confused about our first few lessons in mathematical physics because there were so many new terms/symbols introduced (unlike regular math classes) like the Kronecker delta and Levi-Civita, so my fundamentals before tensor analysis is really fragile. This video really helped me review our past lessons, and I can say I have a much better understanding now on matrices and coordinate transformations. Thank you for being a very good teacher. Your explanations are very clear. Xx

[@shaylasultana7693]

It's great how you find the angles by thinking about how exactly the rotation is happening and not drawing parallel lines to find similar angles

[@account1307]

Straight-forward, helpful, intuitive, and to-the-point explanation - thank you so much

[@unknown360ful]

AWESOME VIDEO! HYPED FOR THIS SERIES! *One suggestion*: Even though I know very little, You should've introduced Einstein Summation in the video first, that would've made your notation a lot less cluttered...

[@swaroopsampad1876]

this series is the best there in the whole youtube .....thanks a lot..

[@clemenstombozichama645]

😭🙌🏾omg I actually love you. Thank you for making my life easier

[@tarunramkanuri3581]

Thank you so much, Andrew.

[@RobManser77]

You’re an excellent teacher. 👍🏻 Thanks for the videos.

[@ImSoCool2403]

I love you for this series of lectures

[@Pillemafia]

This is some good educational shit right here, keep it up!

[@abdullahbinjahed6900]

Rick attitude

[@NovaWarrior77]

Two years later, thanks Mr. Dotson!

[@klong4128]

Good tensor-concept applied to static engineering vector (physics transformation vector) as an introduction to real-tensor (Einstein notation) such that layman easily understand from familiar realm understanding the complicated-unknown-tensor realm .Good example of illustration .

[@vwcanter]

These lectures really good. It's useful to go through the notation slowly.

[@ethanhall1080]

I loved your explanation so far, despite only making it a couple minutes in. While some of the notation looked a bit foreign from your explanation I started to understand what the underlying concepts were. Thanks for these great videos! :D

[@bikrampokhrel5937]

Ethan Hall ticket

[@goncalofreitas2094]

Excelent video! Thank you!

[@petermarshall8212]

Excellent presentation … so helpful!!!

[@crosisbh1451]

I really like how you explain setting up problems. Could you do more videos on setting up problems? My mind is drawing blanks staring at my practice problems in this classical mechanics textbook.

[@pablotnr]

Thanks man! Just started taking math methods this semester... really useful

[@The-Devils-Advocate]

The fact that actual math and physics is actually being taught surprised me, since most (if not all) videos I have seen from this channel are meme/joke videos!

[@ml1186]

amazing! Andrew, thank you :)

[@josedanielbazanmanzano9607]

I just found your channel and woww, just wow

[@hybmnzz2658]

My engineering exams are in a few weeks and this is way too interesting.

[@rajaradi802]

Only physics major with a good handwriting is Andrew dotson

[@miqueltube87]

Congrats for the great videos man!! I found a couple of minor errors at min 38. i) The relationship between phi and theta the way you are expressing them should be: phi = 90 - theta. ii) This error then cancels with a second one here: cos(90+theta) = sin (theta) !! With the correct phi / theta relationship, you can use instead: cos( 90 - theta) = sin (theta)

[@feleous5081]

I love your vids.

[@yonathanmagendzo4110]

sos crack amigo! tus videos me han salvado la vida muchas veces. Si alguna vez estas en New York te invito un cafe.

[@Mforader1792]

Thanks for putting this out dude!!!!

[@Mforader1792]

Sup re runnnnnn!!! And i got books lololol thanks again dude🤣🤙

[@saarahhall1644]

thank you thank you thank you for this my good sir

[@waverly2468]

Tensors were mentioned in "The Outer Limits" episode The Human Factor (1963) which is one reason why I got to be curious about them.

[@ramonfranco1390]

excellent clarification.

[@Yuvrajsingh-sr4ic]

You are love Sir

[@ianprado1488]

Great video!

[@zokalyx]

Andrew. Yeah it's me again lol Just wanted to make a suggestion for the series If videos are long (like this one), it might help to write the list of topics to be covered in the video and write clear titles when you are talking about them. Also, I don't know your exact plans, but I would do these videos once a week; you give some time for your viewers to see the latest video on the series and most importantly, to let the ideas sink in (I feel how tough these topics can be for people with no actual university-level education in physics, such as many viewers here). Anyway, great video :)

[@AndrewDotsonvideos]

Thanks for the feedback! I think a list of topics at the beginning would be helpful. I wouldn't be able to make them more frequent than once a week if I tried. I think once a week or once every other week will be how it ends up playing out. Especially once we actually get to the more subtle math.

[@duncanw9901]

Hint: double speed. A big thumbs up from me Andrew. Taught me more than my hs physics teacher has tbh he's been held back by people bad at physics.

[@EssentialsOfMath]

Duncan W hey you're subbed to me :)

[@ollielast2901]

I tried modern motion and gravity etc and this helped

[@hegaregieara1002]

Oh wow that was a very long video but I made it through and it's defo worth it!

[@jonatanpaschoal7362]

Excelente aula.

[@ahsanrizvi2205]

Extremely useful for me

[@dy_physics9183]

I started this series when you started it and I got the book to follow along with but I hadn't taken linear algebra yet, so once you got to sections where that as a prereq is kinda necessary I stopped. I've long since taken linear and am now taking GR so I can continue where I left off lol.

[@athul_c1375]

I am saying again you are going to become a great Professor

[@architsharma9836]

very nice explanation.

[@LoyalZen0x]

Have my last exam tomorrow but this couldn't wait :)

[@davidsweeney111]

I'm so excited, I'm in too deep

[@HuggumsMcgehee]

In the second example with the rotated (prime) reference frame, I got confused by the fact that you labeled your x-component as negative in the triangle and your y-component as positive which you changed later. If you use your angles properly, which you mostly did, then you don't need to place the negative sign in front of the components. They'll come out positive or negative based on their angles with respect to the relevant unit vector. I had to think about the issue with the sign of cosine as you did it in the problem. You're correct in saying that it's an even function, so if I flip the sign of the argument, the sign of the result will remain the same, however, because the angle between the unit vector i-hat and the gravitational force vector is oblique the resulting cosine will be negative. If you flip the sign of the argument, the cosine will remain negative. That was really throwing me for a minute and it's a very easy mistake for anyone to make. Honestly, still having a hard time thinking about how you introduce that to a student without introducing the sign in there in a potentially confusing way. You would've had to write "cos(90º+θ) = -sin(θ)." I also think it might be helpful to actually draw the unit vector and force vector of interest tail-to-tail. That way you can see what angle you're dealing with better.

[@apoorvsom.5656]

woohoo! I'm happy I landed here.

[@Sabir40]

Starting this Tensor series from today😁😁

[@freddupont3597]

Hard to discern if you've never heard of it at 11:50, it is the Kronecker delta: en.wikipedia.org/wiki/Kronecker_delta (also in the video description, I see it now)

[@beegood5908]

I feel shame being a graduate student and still having a hard time grasping the nature of tensors, hope you will bring me some different perspectives since group theory and representation in quantum mechanics is just other words for nigthmare to me right now! Love from Paris and be safe

[@angelmendez-rivera351]

I think you should have written the Levi-Civita symbol with lower indeces, and the component of the cross product also with a low index, a contravariant symbol. Or, alternatively, kept the upper index on the cross product, but written the Levi-Civita symbol with mixed indeces. This would have been especially useful, since it would have allowed you to seemlessly transition into explaining the Einstein summation convention in later videos in the series. Also, you should have emphasized in the explanation of the transformation of coordinates that φ = 90° - θ, since it would have driven home the point that sin(θ) = cos(φ). There is more to be said, but you already acknowledged it in your edits and comments, so it is all good.

[@pneptun]

this guy's good!

[@proof-xx1vv]

DUUUUDE this is pure gold. I just have a minor issue: could you use barred letters when changing to another coordinate system? the primes look the number one (1). Thanks bro.

[@tomergandelman8083]

life saver

[@andresfrr100]

Hi. In time 38.22 θ+ϕ = 90°, just prolong N 'til it reaches the x axis.

[@subhrajitroy1477]

0:40....Me to myself: "Do I need to decipher these cave paintings?"

[@NovaWarrior77]

Me to myself: why am I so alone???

[@subhrajitroy1477]

@@NovaWarrior77 XD lol

[@sid.x]

Tensor is something that transforms as a tensor

[@muhammadasifzubair4923]

at 39:14 N of x should be positive . cause cos(90+ θ) = - sinθ

[@muhammadasifzubair4923]

same goes for 47:24

[@ahorribleperson3302]

Isn't he still correct since the block is sliding in the negative direction? I think he just skipped a step, otherwise the equation would show that friction + the x component of the normal force is equal to the force, and therefor the acceleration experienced by the block, which is obviously not true. Sorry if I missed something.

[@6900xx]

I think he made a bouble mistake and got it right. Phi is not equal to 90°+theta, instead it should be 90°- theta. So eventually he got it right. I'm currently at 39:11

[@6900xx]

Sorry I meant "double mistake". Typo* :'(

[@cerwe8861]

@@6900xx θ=0→Φ=90° and θ=90°→Φ=180° so Φ=90°+θ

[@compphysgeek]

just wanted to see if there are any comments regarding the whole \phi = 90 deg + \theta and \phi = - 90 deg - \theta and is cos(90 + \theta) = - sin(\theta) or +sin(\theta) caused any confusion and there were some comments were it caused confusion how you dealt with the angles. The whole confusion could have been avoided if you had identified the opposite angle as \theta instead of introducing a new angle \phi. you want to make your life easier that's why you change coordinate systems because equations might have a simpler form in different coordinate systems. you don't want to introduce difficulties by a poor choice of angles ;) .. other than that, great series, keep it up!

[@ashishdahal4671]

Can You give me the links of all the mathematical physics videos? I like the way you explain the mathematics part.

[@tridivsharma2342]

Richard Feynman defo taught you dude, yk how to teach, thanks for this series

[@yuvraj7214]

Hey Andrew, the video is awesome. I had been waiting for it. One small thing though is that the sound is quite feeble... a more sensitive mic or talking a bit louder might help.

[@AndrewDotsonvideos]

Elvie Shane yeah it seemed like the audio was good when I was facing the camera, and then died off once I faced the whiteboard

[@ryangetz8965]

First time I have ever seen the levi-civita symbol in super script. I suppose it doesn't make much a difference but that was my one issue in the video not already mentioned in the comments.

[@chrstfer2452]

The currency comparison is awesome. Edit: but why degrees instead of radians?

[@Trixex]

I found the begging confusing but at the end I understood everything, thank you for this very helpful video

[@raghavdodla1376]

No one's begging here fucker

[@ChrisChoi123]

@@raghavdodla1376 woah calm down young man

[@djalaleddinezemali6075]

Hello Andrew did u make a video about derivation of the energy-momentum tensor of electromagnetic field,if not plz can u make one,and thnk u alot ur videos really help me.

[@HuggumsMcgehee]

At the beginning of the video when you wrote the tensor transformation equation, I had difficulty determining the difference between which symbol is a prime symbol ( ' ), a number one, or the letter i.

[@IntelR]

From The first derivation, if I want to solve the N2L, how do I eliminate the normal force in the equations of motion?

[@juliecrawford9428]

Thank You for super memories, memories, thanks to you, I'm going to resurrect. 😉

[@AndrewDotsonvideos]

memories of tensor calc?

[@juliecrawford9428]

Andrew Dotson memories of the fabulous training that mathematics provided a poly-major. Calc. is only the beginning.

[@juliecrawford9428]

Andrew Dotson Once you've left linear and diff., the real fun starts in upper division. To answer your question, I want to return. You really cannot understand a great deal of academics, with- out an armload of mathematics. And I am speaking about a great deal else aside from math. I had so very many diff- erent interests that I only later discovered were so much more fulfilling when the hard-biting conceptualization of 'out-side-of the-box thought processing came my way. Regardless of whatever your interest, don't poo-poo math and phil. sub disciplines. Believe me, you'll thank your- self later.

[@gabriellove4361]

Is the Levi Civita symbol related to the sign of a permutation used when you define determinants and stuff?

[@FernandoVinny]

Your beard is so perfect man

[@coffeeguy.3438]

Hi. Just wanna ask if the series can prepare me well to dive in Sean Caroll’s GR textbook.

[@urosgrandovec3409]

i would just like to know if the directional derivative tells us how much the function is changing in the direction of the normal vector? the gradient itself tells us the direction in which it is changing the most.

[@AkamiChannel]

If someone used a left-handed coordinate system, would it change the formula for how to calculate the determinant?

[@alexschopbarteld922]

Hey, i have a question i really like your videos and am interested in math, in what order should i learn math, like in highschool it is basic algebra, trigonometry, taking derivatives, doing integrals ect. how would you approach university level math, diffirential equations ect.

[@AndrewDotsonvideos]

Calc I, Calc II, Calc III, Differential Equations / Linear Algebra, Partial Differential Equations. And maybe throw complex variables in there somewhere if you're feeling froggy

[@MrGold-17]

"So you're going through all of this physics and then you see *F=ma* and you're like 'wait, really? since when?'" haha lmao

[@compphysgeek]

just a small remark since you said that a hat almost always denotes a unit vector and you left out the hat when referring to non-unit vectors. the e - notation for vectors, i.e. e_x, e_y, e_z, comes from the word Einheitsvektor, meaning unit vector.

[@prashantdahiya711]

Divergence should diverge, so it must be coming from a source rather than going towards a sink.

[@pavelpivaev]

Hey, Andrew! Nice stuff. But I got stuck with one point concerning a unit vector's length. You said at around 18:50 like let's consider basis vectors but with a length not equal to one. What does it mean to a basis vector to have a length different from the unit one? I thought that we measure lengths relative to basis vectors, lengths of which are conveniently thought of as the unit ones. Then relative to what do we actually measure the length of a basis vector?

[@number1RabidPlatypus]

What's important here is that the basis vectors are not of equal magnitude. Sure you could call the magnitude of one basis vector "1" for convenience but the other basis vector(s) would not also equal one. So relative to any arbitrary magnitude both basis vectors can not simultaneously equal 1 for any units of measurement you could choose.

[@nahimafing]

can you do a video on 4-gradient please!

[@peterclark5244]

Man, where was this when I took Physics 1?

[@jacobharris5894]

You won’t see some of this stuff in physics 1 but you will see it in Physics 2 or calculus 3.

[@ModuliOfRiemannSurfaces]

Are you going to be covering forms as well?

[@steinraf]

That ampersand though

[@GojoSenpai25]

Minor change: use x hat, y hat, z hat notation because in engineering and physics they are always used instead of the mathematical notation

[@AndrewDotsonvideos]

DocAlex personally I use xhat yhat zhat notation myself. I was just sticking with notation used in the book

[@angelmendez-rivera351]

DocAlex That's not entirely true. I've used plenty of physics textbooks that use i, j, k. My professors also used that notation, and it even gets used in papers.

[@federicopagano6590]

17:40 isn't that the covariant coordinate definition? Why the upper index?

[@yevgeniygorbachev5152]

Does the Levi-Civita symbol work for non-orthonormal bases?

[@chimetimepaprika]

I like when he calls me a smart person even though my intelligence is average. :)

[@mohtasimtamjeed]

*fix it in post