The CURL of a 3D vector field // Vector Calculus

Рет қаралды 59,580

Күн бұрын

One property of a three dimensional vector field is called the CURL, and it measures the degree to which the field induces spinning in some plane. This is a local property, which means there could be a different curl at each point, in contrast to a more global property like the circulation around a larger curve. We've actually seen the "kth component of curl" previously when we were talking about the analogous concept in two dimensions, and then we called it circulation density. In this video we will also look at the del operator which provides a quick way to compute the curl of a vector field. We finally note the curl of a gradient field is zero, and connect this back to the notion of a conservative vector field.
0:00 Definition of Curl
0:55 Geometric Meaning in 2D
2:16 Geometric Meaning in 3D
4:02 del operator formula
5:55 Curl of Gradient
6:43 Test for Conservative
MY VECTOR CALCULUS PLAYLIST:
►VECTOR CALCULUS (Calc IV) • Calculus IV: Vector Ca...
OTHER COURSE PLAYLISTS:
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS: • How to solve ODEs with...
OTHER PLAYLISTS:
► Learning Math Series
• 5 Tips To Make Math Pr...
►Cool Math Series:
• Cool Math Series
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com/shop/treforbazett
SOCIALS:
►Twitter (math based): / treforbazett
►Instagram (photography based): / treforphotography

Пікірлер: 59

@mythic3187 3 жыл бұрын

I just want to get my computation correct , but at your 6:20 mark your k-hat has Fyz-Fzy. I got Fyx-Fxy. Did I miss something?..Thanks Doc

@DrTrefor 3 жыл бұрын

Oops, nope k-hat is [f_yx-fxy], I did some weird symbol permuting...thank you for catching that!

@mythic3187 3 жыл бұрын

Ok , awesome . No problem Dr. Bazett

@Dragon-Slay3r Жыл бұрын

F is in the curl thanks. Aka The missing body

@Dragon-Slay3r Жыл бұрын

It's a F hat turned 2 10 o'clock position

@angeldude101 Жыл бұрын

Well, Fyx and Fxy inherently both have units of x̂ŷ or ŷx̂ depending on convention, and Fyz and Fzy both have units of ŷẑ or ẑŷ depending on convention. When written out explicitly, both the minus signs and which component they fit in become trivial. (Flipping the order of the two directions always introduces a minus sign.) And yes, choosing x̂ŷ or ŷx̂ is _purely_ a matter of convention. I think the traditional cross product uses ŷx̂ which seems rather _sinister_ if you ask me. Note: "sinister" means "left-handed"

@SHREYANAND-dn6jc Күн бұрын

You have been a blessing to students you want to visualise vector analysis aspects. Thanks a lot!

@robphippen 3 күн бұрын

Love the 'spinner' as a way to elucidate what curl means. Excellent.

@garmands 3 жыл бұрын

I am binge watching all these vector calculus series. All of this is in the calculus book, but this is just so much better because of the way it is presented.

@DrTrefor 3 жыл бұрын

Thank you. Have fun and make sure to comment when you make it to the end!

@baberkhanbk Ай бұрын

The curl equations for a vector are definitely harder to remember than divergence, laplacian and gradient ones. Thanks for explaining the Curl of a vector.

@drover7476 2 жыл бұрын

Your videos are so brilliant. I honestly find them so helpful in giving me a true fundamental understanding of the underlying physics and maths I am studying at university. Thank you mate.

@euyin77 5 ай бұрын

I have taught vector calculus since 2003. Today was the first time someone asked me about the physical and geometrical interpretation of curl. And it was the first time I had to think about it! And the answer I gave him was exactly this. Before watching this video! But I used a ball with an initial velocity that returned to the same point.

@adityaprasad465 2 жыл бұрын

At 3:51 "That axis is the curl." May be helpful to point out that the axis is the _direction_ of the curl. The curl's magnitude also encodes information (about how fast the paddle would be spinning).

@zl4518 3 жыл бұрын

It finally clicked! This video leads perfectly to Stoke's Theorem, and reveals how it relates with the Green's Theorem. After all the thinking and struggling, thank you!

@DrTrefor 3 жыл бұрын

Glad you liked it!

@crazykidgaming Жыл бұрын

I enjoy understanding the concepts behind math; this really helped. My teacher is good about teaching theory and computation, but I’ve been zoning out in class so this was a great supplement!

@pankajjkumar369 3 жыл бұрын

Thanks sir for explaining in easiest way.

@alexmassy 2 ай бұрын

So sick ! Love the clear explanation. It gives me great intuition ! Thx

@tonystank1309 3 жыл бұрын

Totally Irrespective of the video content... But just saying.... Sir, I had my Probability Exam yesterday and the night before it I've seen your video on Markov Chains and Transition Matrix (Liked it, amazing content and easy )... I had the same question asked during the exam, and boi ohh boi... I've written it around 2 pages... Kinda wrote your whole video's transcript!! Hope he doesn't find it out...😆😎... Haha Anyways... Good work by you Sir, thanks for making concepts much more intuitively understandable.

@DrTrefor 3 жыл бұрын

I'm so glad that was helpful!

@ndydiy9963 2 жыл бұрын

Another great video

@bharvichavda3725 3 жыл бұрын

Just so gr8 series that it never ceases to amaze me. Thank you so much Dr. Trefor.

@discussionfortherightway.9725 3 жыл бұрын

Great Video Doc.👌 I was introduced to the Gradient sign as Nabla and to the Partial Derivative Sign as Del. All the different conventions are a bit confusing... Have a great day!

@thilinadisanayaka7382 3 жыл бұрын

great explanation through the full course

@DrTrefor 3 жыл бұрын

Glad to hear that!

@getshrekt9023 2 ай бұрын

Great video 🙏

@HosRo4161 Жыл бұрын

Excellent!

@jesushernanjimenez6471 2 жыл бұрын

Thank you!

@marksmod 2 жыл бұрын

Don't forget that the cross product can be expressed as the "formal" (which, according to Wikipedia, means that it just is that way because it fits, without further justification) determinant, so the cross product at 5:10 is more of a mnemonic aid. I think. The other thing that has been bugging me while learning this stuff is that nabla is a function which takes functions as its first argument, but this is obviously not being done in the cross product. Since the cross product multiplies the elements together, you end up with something like nabla(multiply(void, stuff)) or multiply(nabla(void), stuff) which turns into nabla(stuff). I guess this sort of similar to how we use Leibnitz notation a bit more loosely like fractions?

@j.o.5957 3 жыл бұрын

OOoo, now we're getting to the good part. Nice. To summarize, the cross product is the important thing to remember. Also, I tried to explain this to myself, that the third one (dN/dx - dM/dy)*k_hat is the curl around the normal to x X y. The same applies to the others. The magnitude of the curl would be sqrt( variable^2 + (...)), which is cool. Question to self: What was the other formula that was relevant for conservativeness? Ah, it was finding the potential function. Nice. I'm getting stoked about stoke's.

@Doivid_ 2 жыл бұрын

Now, I feel the need to point out that people who are watching this video should know what an operator is ahead of time. Like if they don't know that then they're watching the wrong video lol

@kudzanainyandoro8241 Жыл бұрын

Divergence goes along with dot product and curl goes along with cross product

@angeldude101 Жыл бұрын

And if you add both together, you get a quaternion. (I'm not even joking, the derivative of a 3D vector field really is a quaternion.)

@krishnaks4622 10 ай бұрын

thankyou

@Dragon-Slay3r Жыл бұрын

The curl is a horizontal rectangle on the tail on the scorpion sting while all of it is verticle more graves

@michaelsanchez7798 Жыл бұрын

What Calculus book do you use in your courses?

@ductanle2806 3 жыл бұрын

Sir you should really do a video on the geometrical meaning of second derivatives, mixed or unmixed. Otherwise great videos!

@carultch Жыл бұрын

The geometric meaning of second partial derivatives is still curvature, like it is for single variable second derivatives. Technically, there's a lot more to curvature than just the second derivative, but for negligible slopes, it does equal the curvature. The unmixed second derivatives describe the curvature as seen when projected onto one of the vertical planes. Project onto the x-z plane, and you see d^2z/dx^2 describe its curvature. Project onto the y-z plane, and you see d^2z/dy^2 describe its curvature. The mixed second partial derivatives are much more complicated to describe geometrically, but the idea is that they rotate the principal plane of curvature away from the coordinate planes. When the mixed derivatives are zero, the maximum and minimum curvatures both occur on either the y-z plane, or x-z plane. But when there is a mixed derivative, it can twist the function so that it could have its maximum curvature occur on a diagonal plane.

@yizhang7027 3 жыл бұрын

Is del a real vector? or is it just in the form of a vector so we can do cross product with it?

@alevelsos 11 ай бұрын

5:43 If we are calculating the cross product shouldn't the second j cap term be negetive?

@carultch 6 ай бұрын

Depends on which method of taking the determinant is. I prefer the rule of Sarrus, where you add up the three products along the positive diagonals, and subtract the three products along the negative diagonals. The positive diagonals are the down-right diagonals. Here, you don't need to think about the negative sign on the j-hat term, because it's already accounted-for. The Rule of Sarrus only applies to 3x3 matrices, so it doesn't extend to the 4x4 case. The sub-determinant method is what applies in that case, and all the even terms along the top row, get a negative sign. If there were an ℓ-hat term in a cross product, it would have this negative sign as well.

@ricardobautista-garcia8492 3 жыл бұрын

First. Finally. Thank you.

@DrTrefor 3 жыл бұрын

haha nice!

@federicopagano6590 Жыл бұрын

If RotF(P)=(2;3;1) is it curled? How much?

@Dragon-Slay3r Жыл бұрын

4 back to front

@sebjornsprauten1406 3 жыл бұрын

This magnificent bastard!

@kazadrix01 Жыл бұрын

isn't j^ supposed to be negative, as in i^-j^+k^ when you compute your cross product? since this matrix has + - + alternation.

@carultch Жыл бұрын

Not the way he wrote it. Had he written [dP/dx - dM/dz]*jhat, then he'd need a negative sign in front of it. When you derive it through sub-determinants, then you do have a negative sign on every even term. When you derive it by constructing the 3x3 determinant and multiplying along positive diagonals and negative diagonals, you don't need to think about a negative sign on jhat.

@angeldude101 Жыл бұрын

Ultimately, it's a matter of convention and doesn't actually matter. If you're explicit with whether you're using x̂ŷ or ŷx̂, then you don't need to remember where to put the minus signs.

@TadStaniels 3 ай бұрын

He went back to the video to see what had been recorded and was shocked at what he saw.

@zoltansafran8 3 жыл бұрын

It is not true (last sentence of the video), that if the curl of F is zero, then the vector field is conservative. The statment is only true the otherway around, take F =(-y/(x^2+y^2)) *î+(x/(x^2+y^2))*j as an example. The curl is zero, but if you compute the line intergral around the center, you‘ll get 2pi. Top playlist though, love your vids! Keep up the good work :)

@DrTrefor 3 жыл бұрын

The statement is if and only if when you have the additional assumption of smoothness, I see I neglected to specify that here, but this was covered in more detail earlier in the playlist:)

@shirinakther5461 Жыл бұрын

At which class people study this???

@ogreeni Жыл бұрын

Usually first or second year of college in the U.S.

@paolovilla717 2 жыл бұрын

You simply didn't explain how curl formula is obtained

@carultch Жыл бұрын

It's defined that way.

@hubenbu 2 жыл бұрын

My problem, the computation is totally trivial, but everything is meaningless, lol, do I need to read a Physics book or something? Thank you! edit: Great illustration, the hurricane-like graph makes sense when I watch it for the second time.

@carultch Жыл бұрын

Yes. An application of curl, is Maxwell's equations for electromagnetism. Ampere's law and Faraday's law of induction, both involve curl, as electric fields interact with magnetic fields and vice-versa. There are also applications of curl and divergence in the Navier-Stokes equations of fluid mechanics.