This Downward Pointing Triangle Means Grad Div and Curl in Vector Calculus (Nabla / Del) by Parth G

Gradient, Divergence, and Curl are extremely useful operators in the field of Vector Calculus. In this video, we'll be trying to get an intuitive understanding of what they represent, as visually as possible.
Hey everyone, in this video I wanted to put a physicist's spin on a heavy mathematical topic. Gradient, Divergence, and Curl are used in many different topics in physics. For example, we see them in the study of gravitation, electromagnetism, relativity, and many more.
The del or nabla operator, represented by a downward-pointing triangle, can be thought of as a vector of partial derivatives. In three dimensions, our vector consists of partial d/dx, partial d/dy, and partial d/dz. What each one of these represents is the rate of change of any given quantity, to which we apply del, in the direction x, y, or z. The partial derivative simply represents the fact that we imagine everything in any other direction to be constant. So if we are trying to find partial d/dx, then we assume our quantity does not change in y and z, and so on.
This vector can be applied directly to a scalar field in order to find its so called "gradient", often shortened to "grad". Now a scalar field is basically just any region of space (whether real or abstract) that can be assigned some number or quantity. For example, on a map we may see numbers or contours representing the height above sea level of each point. This is a scalar field, because each point can be given a number - its height above sea level. And when we apply the gradient operator to our scalar field, what we get in return is a vector field. This vector field represents the rate of fastest change of the original scalar field at every point. In other words, for our map analogy, at every point the gradient points up the steepest climb adjacent to it. And each vector's size represents exactly how steep that climb is. (By the way, a vector field is just a region of space in which we can assign a vector to each point - size and direction).
Additionally, we can apply the nabla operator to a vector field, if instead of direct application, we choose to take the dot product between our del and the field. Now a dot product, or a scalar product, between two vectors, simply consists of multiplying corresponding components of each vector and then adding up these products. If our vector field describes the electric field generated in space by nearby charges, then taking the dot product between the del and our field, gives us what's known as the "divergence" of the field. This is often shortened to "div", and tells us exactly how much of the field is being emitted or absorbed by each point in our region of space. In other words, it is a measure of exactly how much each point is a source or a sink of the field. And as it turns out, only points where there are charges, can be sources or sinks. Positive charges are sources of the electric field, the field seems to emanate from them, and negative charges are sinks because the field lines end there. This is determined by one of Maxwell's Equations. So basically, finding the divergence of a vector field results in a scalar field.
Finally, we can also find the cross product between the del operator and a vector field. A cross product, or vector product, usually refers to a measure of alignment between two ordinary vectors. The end result is a third vector, perpendicular to both the originals, and this vector will be as long as possible if the two originals are exactly at right angles to each other. But if they are aligned or anti-aligned, then the resultant vector will have a length of zero. However, taking the cross product between the nabla and a scalar field measures the "circulation" of the original field. At each point, we find a vector that represents how much a little stick / piece of plastic would rotate if placed at that point in the field. If our field represented the flow of water in a lake, then we could imagine putting a stick into it and in some regions it would spin. In these regions, we could represent the "curl" of the water flow field as another vector, pointing along the axis of our stick's rotation. And the more the stick rotates, the larger the vector. If it rotates clockwise, the vector points downward, and if it rotates anticlockwise, the vector points upwards.
Timestamps
0:55 - Nabla / Del and Partial Derivatives
3:21 - Scalar Fields and Gradient
5:08 - Vector Fields and Divergence
8:50 - Curl
11:04 - Applications (in Physics)
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[@ParthGChannel]

Hi friends, thanks so much for watching this video! Now that you know about Grad Div and Curl, check out my Maxwell Equation playlist if you'd like to see how they're applied in the world of physics! kzbin.info/aero/PLOlz9q28K2e6aNgl1zt1xccyy4Ofl3YAk

[@snehapatel8292]

Hi Parth, great video. Can you please discuss about cooper pairs in superconductors in your next video? Thanks

[@shama_k2604]

Please cover the remaining equations so that you explain all 4 of them in both their integral and differential form...!!! It's the need of the hour Parth 🙏🥶

[@BlueGiant69202]

This video is very good but does not make use of Geometric Calculus, so will make movement to 4d and the Dirac equation a bit harder. Would you consider presenting and explaining the Maxwell equation of Geometric Calculus? The symbol del/nabla is used in Geometric Calculus for a similar but more general operator ( a box is sometimes used to make it clear that it is not a 3d Del but the Dirac operator acting as a derivative of a point on a higher dimensional manifold. Del F = J is the Maxwell equation. Del F = Del dot F + Del wedge F. F is a BIVECTOR field. Anthony Lasenby on use of Geometric Algebra and Geometric Calculus in Physics kzbin.info/www/bejne/rmjIfXiqotZphrs

[@kingbeauregard]

In 1986 or so, my differential equations instructor (Dr Joseph Egar, Cleveland State University) defined operators thusly: "An operator is a animal that eats functions and spits out functions." Then sometimes he'd correct himself: "well, not so much spit them out, they come out the other end".

[@thermrm]

Lmao

[@Sasukej2004]

idk why but lately ive been seeing a lot of physics and math comments from "1986" talking about their experiences. Can sum1 pls tell me whats going on?

[@darrellrees4371]

@@Sasukej2004 - I honestly could not begin to guess why that is. But, coincidently enough, in 1986 I was studying math and physics at university so.....

[@sugandhpriya6734]

Hi Parth. I really like the way you explain difficult concepts in a very easy and relatable way. Could you please make a video on tensors.

[@dineshkumarv9493]

😂

[@_..---]

This channel is amazing, you always find a way to explain things better than my professors.

[@_..---]

​@ཀཱ no thanks, it didn't support xroach last time I checked

[@HackersSun]

Thank you I thought that I was the only one 😀

[@SALESENGLISH2020]

It's great that Parth Explains the Intuitive meaning of these operators. We use these in fluid mechanics. It is hard for most students to understand what these operators really do. Keep it up!

[@mwafrikahalisi2549]

I would rather say it's hard for most teachers to explain how these mathematical symbols does represent the concepts more intuitively. Generally, students will understand the meaning of these operators if they are clearly defined in their simplicity as in this series of videos by Professor Parth G. I am in awe at the simplicity of which Professor Rarth G explains these seemingly complex concepts. Kudos to the good Professor!!

[@dakshsharma5661]

you make me love physics.

[@pragyanshsingh6093]

This is one of the best video explanation, I have watched on the vector calculus. Thanks a lot.

[@jas.c4986]

The best explaination on KZbin!!! Finally not confused anymore.

[@charlesudekwe2404]

Honestly, I love these explanations, they are very intuitive and brings out the physical meaning of these concepts

[@BitwiseMobile]

Watching these makes me want to freshen up on my calculus. It's been almost 30 years. I do remember curl and some of vector calculus, but it's foggy and I need a refresher :)

[@KsmithBlackstar]

This is EXACTLY what I needed. You explained everything I wanted to know but didn’t know how to ask.

[@user-bz7ct3iu3v]

You are an outstanding teacher

[@ParthGChannel]

Thank you so much

[@ekt2656]

I’ve been subscribed for a couple weeks now because someone mentioned the Lagrangian to me and oh boy I’m staying subscribed

[@isd99]

Vector calculus love it

[@aelolul]

Back in college I was taught how to calculate these things, but never properly shown what they actually do (or didn't study properly to grasp it on my own). Thank you for this video, it was great.

[@penumbraman99]

Wow! I’m going to have to watch this video clip multiple times. There is a lot info in this one. I’ll have to go back and rewatch your Maxwell equations video clips, too. I always wanted to understand Maxwell’s equations. I’m putting this video on my favorite list now.

[@cadence_wav2423]

Just a few realize how amazing is the musical taste ,which Parth G has 🤘🏻

[@thecaribbeanbookworm5066]

This channel just keeps getting better and better. I love this.

[@halaanbar-ko1ez]

i've watched this video more than once , every time i'm amazed , you explain it so simply like telling a story . thankyou

[@JeremyChung]

Holy crap. I'm only a first year eng. student and this made perfect sense. Super well made and insightful

[@chanlee5310]

nothing else you are just creating scients in this era using your videos dont stop this keep going just damn great contents

[@Kracheta]

Wow, what an excellent video! Thanks so much for explaining so clearly. I found particularly helpful that you give the “types” of each operator, saying what kind of field they operate on and what kind of field they produce!

[@LucasFerreira-dv6ol]

Dude! That is insanely amazing! Your didactic is incredible and the illustrations are so helpful! Thank you for this content, I really appreciate it!

[@placeboantwerp4312]

Hi Parth, recently discovered your vids and am binging on them at the moment. They are all fantastic, in particular skimming the maths in favour of focusing on meaning. I really think you get that balance just right, for me anyway. This one is my favorite so far. Have loved to try and visualise Grad, Div and Curl for decades now and this was the best explanation I've come across. I'm going to try and give it a run with my wife, who doesn't seem to have my (our) love of physics. I think Grad, Div and Curl can all be explained with a swiming pool and velocity (speed) for vector (scalar) fields, so will supliment your video with my explanation of that. Keep up the fantastic work mate!

[@suparnaprasad8187]

Oh my god, this is a gem of a video. Today was my first physics class in undergrad and I did not understand head or tail of this whole concept but your animations and explanations have made it crystal clear now! Thank you so so much! This really made my day!

[@raphaelkaume]

My friend you have such a gift for exposition please never stop. I'm in awe. Have watched just a few videos but immediately sub'd. In fact they're so good I've had to pause them at times and go make a nice cuppa coffee and come back later to enjoy with pad and pencil. Why oh why didn't we have teachers like this in school!!? Can you imagine on the Internet with comedy videos and cute animal videos yet you're making us chose to return again and again. Pure talent man you're opening new worlds for me. Now one of my top 4 sites for Science in general and top 2 for physics. Thank you

[@roygalaasen]

Just dropped by to encourage you to continue doing this. Your explanation is simply amazing. Thank you.

[@shaharudinhamidun2117]

You always be able to explain the complex mathematic and physics in easy way so everybody could understand the subjects. Really enjoy watching your channel.

[@daveuk1324]

I love this guy! Wish he'd been my physics prof back in the day!

[@andyk2181]

Excellent video, wish I had found such a clear explanation of these concepts earlier. When studying electrical engineering I found it confusing that magnetic fields didn't have sources and sinks like electric fields. Actually, now it makes sense because magnetic fields don't really exist (at least that's my thinking), it's merely a relativistic effect of moving electric charges. I mean, it seems even space has sources and sinks.

[@leoclarkin5944]

Excellent thank you, One of my Covid projects was to have a thorough understanding of Maxwell's Equations, which I last studied at Uni, 40 years ago! Back then, I took a semester course and got a very good result, mainly because I understood the Math (i.e., the Del Op gradient, Div and curl. Now the physics I still. after the course, I did not have a clue on the meaning of Maxwell's equations; now with your review of the Grad, Div and Curl I'll attempt to have a deep understanding of the ME ....Let there be Light! Again Parth much thx

[@curtpiazza1688]

Wow! Clear and concise presentation! Thank you so much!!!

[@EpicMathTime]

Beautiful presentation, as usual.

[@biniyamzegeye3697]

man, this man makes it look much easier

[@cleitevieira]

I wish my teachers during my Physics years had taught me operators like this... Thanks, Parth!

[@guilhermemelo4434]

It's funny, I subscribed to your channel 3 weeks ago, being curious about the Lagrange mechanics. So, you posted a video about virtual images, 2 days before I get to this subject on my course, and now you posted a video about these operators, on the same day I got to Maxwell Equations hahaha

[@briann10]

eyyy same here, started with lagrange few weeks ago too.

[@yassinesafraoui]

I have a feeling that this guy is teaching the course you're studying hhhh

[@edubezerra35]

@@yassinesafraoui Or maybe some teachers are stalking our youtuber :)

[@racpa5]

An extremely clear explanation. You sure talk about Maths gooder than a lot of other people.

[@chyldstudios]

Clear explanation of these concepts. Well done.

[@overlordprincekhan]

Such a clear explanation! Keep up the good work bro!

[@malefetsanekoalane4549]

Outstanding explanation as always.Thanks a lot Parth.

[@mohamedah9133]

You are teach us better than our doctor 👨‍⚕️ of physics

[@ashfaqahmed9574]

Perfect explanation! Thanks so much!

[@andyrechenberg]

Best explanation of this topic that I've encountered. Great work! Cheers from Houston, TX.

[@rd-tk6js]

very insightful yet concise, thanks !

[@timhowell6929]

Awesome vid Parth, thank you!

[@gustiuwah]

I'll be writing an exam about this topic soon and this video couldn't have come at a better time. You da best Parth!

[@robineve2812]

Thank you for your videos explaining formulae. I especially like your analogies drawn with real life phenomena, like vectors as wind charts, and, in another video, divergence compared to running and draining a bath. All these make understanding easier. Thank you

[@iestynjones5796]

Brilliant video. Thank you. You are a gifted teacher.

[@issolomissolom3589]

One of the best and most concise explanations ever Thank you so much Why youtube algorithm doesnt show excellent videos like this Liked and subscribed

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video. I never thought I'd see such an informative, yet concise explanation of these topics.

[@ferhatnusreturuc1809]

Perfect explanation, perfect speech, thanks, it is good for us to watch your good job.

[@JB-123]

Amazing video. Solid explanations, thank you.

[@Wordsforexistenxe1911]

Sir you are doing a great job... Curious minds are happy now Thank you sir

[@KetanSingh]

You're a great teacher Parth

[@shankarim8432]

i was searching a legend like you to explain things in a smart wayyyy... since 2020

[@souptemba3092]

thank you, this is perfect. exactly what i needed.

[@nabrajpanthi9857]

keep it up...we need more of them to make education exciting!

[@mahanpourfakhr2267]

This was beautifully explained

[@richardngjoobeng1403]

This video is AWESOME ! Well Done !!!

[@muddysneakers1]

this is AMAZING! thank you for a great video

[@60pluscrazy]

Fantastic explanation 👌

[@terencewright2223]

Excellent video,thank you very much. Your videos or so well done and enjoyable.

[@MohdIrfanZ7]

Great effort at explanation.thanks

[@patrickbrenier4361]

bravo man, you are truly gifted for teaching physics! sharing with my son;-)

[@monoamiga]

Dude! You're beyond awesome. I watch physics and math videos for years (for the sake of fun and curiosity - I'm not a professional scientist, I'm a professional musician ;) ) and I FINALLY got it. This was my "aha" moment and I actually found these equations intuitively simple thanks to your straightforward explanation!!! Thank you!!!

[@jorgeromeu]

Thanks a lot, im taking a numerical methods for PDEs course and i dont have the background in vector calculus, this video helped me a lot

[@raorahulyadav4794]

amazing and clear explanation

[@pratikagarwal6763]

lovely explanation

[@nekelly123]

Great explanation. Thanks!

[@AZ-hj8ym]

This is so beautiful！！I think I got it.Thank you!

[@gavinkanowitz7001]

Great video. Excellent resource for Physics teachers such as myself.

[@nadiakhalili9412]

the way you explain it makes it so attractive:)

[@mikeg2798]

Wish I had had that succinct explanation at university. Nice video.

[@Astartos22]

Brilliant video, thank you

[@BerkKara]

Nice and clean explanation. Billuruna saglik.

[@maths-pro-by-khan-sir]

THANKS A LOT SIR, FOR SUCH NICE EXPLANATION 🙏🙏

[@highbee700]

Thanks Parth G

[@erahmawati]

excellent explanation 👍👍👍👍

[@dennisbrown5313]

Good vid; I especially like your first def. of the Del. operator on a scalar field. In your discussion of curl, this would be better served by using geometric algebra instead of the std. curl. The "outer" product makes far more sense then the cross product.

[@rukeshveera5608]

thanks for great explanation

[@TheRealJavahead]

Whoa! Great video. Why have I only just now found your channel? Subscribed.

[@dylenweerasekara9533]

Thanks, For introducing vector Calculus.

[@jones1351]

Excellent lesson.

[@jannegrey593]

I think we will need that last Maxwell Equation! Great video as usual.

[@rakeshgaddala]

Very useful to understand

[@TechSouls-wn6jc]

Nice explanation

[@killaken2000]

I recommend the books * Grad div curl and all that by schey * Student's guide to maxwell's equations by Fleisch They helped me when I was getting my EE degree

[@ashfakahamed3451]

this is amazing . Sir I don't know how to say thank you to you......

[@Hasnainkhan-qd3wm]

Wawooooo Amazing Job Bro 👍. Very short time to all my questions is clear.... Thank you so much 🥰💖

[@alameenabdullahi4812]

I really enjoyed and found the video helping

[@marlovsk1]

Great work

[@rd-tk6js]

Intuitive explanations !

[@Verschlungen]

As always, Parth G's presentation is lyrically beautiful, concise, swift, and very informative! (As a point of comparison, I'm thinking of the "classic" by H.M. Shey entitled "div, grad, curl, and all that" [sic], with its odd sequence [Chapter II div, Chapter III curl, Chapter IV grad] and its bizarre notation explained on page 4n1; ugh.)

[@rahulr9301]

WOW in the middle of your video i paused and put like and subscribe and then i resume:). Rocking bro

[@parashar1505]

if maths was taught like this in schools, we all would be mathemticians now! genius!

[@stevekru6518]

If math teachers understood the math then it would be taught that way.

[@mahanteshshettar2318]

Hi parth , It was a great video.... I learnt the practical way of thinking how this operator works..... Request : Do a video on ORBITAL MECHANICS , or Equations related to Space science and rocketry......

[@syte_cloud9829]

this is really useful man thank you!

[@steves3948]

Your videos are amazing.

[@maxgold6383]

That was awesome. Thanks.

[@032_pranjalagarwal9]

Some day ago i read in my undergrad course, pleasure experience.

[@tombouie]

Quite well-done