The reason behind this channel being underrated is..that most of the students just study for marks not for the concepts (schools has made them like this) .....there are very few people who look for intuition of the concept...and sir u are a blessing for us...love from india🇮🇳
I agree. If I dont understand or AT LEAST have an idea what is happening behind all of these math formulas, I feel like I am not really learning nor understanding the topics
For me a physics student, this channel I just found is a goldmine... You can´t imagine how poorly math is lectured by our professors. Thank you and all the other great youtube channels like 3blue1brown!!
@@DrTrefor Hahaha your way of explaining these concepts is just like how a good physicist would do it. You really emphasize on intuition and getting a grasp of the conceptual idea, before you dive into the math (aka the toolbox of a physicist hehehe). Also a physics student here😁. Before seeing this comment I was surprised that you explain these concepts in the exact manner and series as our books are, but with even better intuition of course.😁 You have really helped me better understand these topics and prepare for exams. Thank you a lot! I hope you continue your excellent job, so as to help even more people (and maybe me again hehe)
Undoubtedly the best educational math channel on RU-vid. I finally understand the intuition behind all of formulas in my calc lectures, makes it a million times more interesting (and MUCH easier to remember)! Thanks for the amazing content!
The moment when you mentioned the relationship between Integration as the area calculation and yet determining something which is just confined to the boundary kinda made me pause the video and think for a few minutes! A hell of an insight there.
Best explanation of all RU-vid videos on circulation in a very small area. congrats. After this video, line integral concept is much easier. You articulate well and presentation sequence is very logical and understandable.
I remember watching this for the first time during my calc 3 time. I hadn't seen a more perfect and easy to understand explanation than this. Being able to visualize calculus makes it so much more fun. Coming back and rewatching it now makes it all nostalgic. Thank you Dr Bazett!
The last misconception mentioned in the video was totally my confusion! Thx for solving this problem, and now Im really clear whats green theorem is talking about! Great video!
These are the best math videos on the internet. Very good for studying for math exams. I'd be happy though if there was a good stochastics lecture for undergrad.
It makes sense how the middle circulation impacts the outer. Compare it to water moving in a circle, if you begin stirring in the opposite direction inside the circle, if would affect the inner flow. Question for myself: The left part of the equation is the circulation around the edge, while the right is the circulation in the middle (as well as on the edge). Why are they the same? Must be because it's not circulation in the middle, but circulation density, which is how much it circulates in a given area. Times it by the size of that area and you only get the circulation. The definition of circulation is "The amount of force that pushes along a closed boundary or path". It's the total 'push' you get when going along a path, such as a circle. So by computing all the small spinning propellers inside an area, you can find the force that's exerted at only the edge of that area. I assume the same way you could change the area, and through knowing the circulation density, you could predict the force needed to go through that line. Thus, is you know the circulation density anywhere, you can calculate the force needed to transverse any simple and closed path.
I have to say, this was absolutely amazing!!! That last connection to FTC at the end was so beautiful I could've cried; that connection between activity at the boundary and inside the boundary seemed a bit less abstract than before. One question: Since this is a double integral with a function of x and y inside the integrand, does that mean that we are technically doing a volume integral? Or, even if we are, would we really be interpreting that number that we get as a volume? Thank you!
Thanks, that was a nice video lecture-but the unintuitive scenario you describe at the end begs me to try and 'disprove' it by way of a counterexample. When I can't disprove it, I'll be satisfied. To the whiteboard!
I'm currently doing my PhD and deal with Stokes' theorem a lot. Particularly using partial integration and product rule on Stokes' theorem to regularize certain singular integrals in Boundary Element Method. Would love for some discussion and exchanges with you :)
Fantastic Video! Content is top notch. Audio does seem to be clipping a bit, if you can try set your gain on your mic down just a touch. Your voice is too enthusiastic your mic can't handle it :D
4:57 how does the single sum change to a double sum ?? any clarity on that please ? it wasnt covered in the video. so you have delta x and delta y but just one sum for i. dont we need a j as well to make it into a double integral ?
but in green circulation theorm- when we integrate sum of all curls on dA... (curlF). k̂ dA....then dot product of two perpendicular vector should be 0 i.e (curlF). k̂ should be =0??
haha nice! I'm hoping the vids coming out in about a week or two will be best. Mic/Camera/Room insulation/Post-Processing all finally on point. People will still complain I"m sure:D
Hello sir I have a doubt .... I understood that circulation density of a vector field and that we can split up a curve into multiple curves(which are rectangles in this case) but at the boundary of the curve we can never truly overlap the curve using rectangles......in standard integration I understood that error shrinks to zero but in this case we are calculating the line integral so I dont understand how the error here shrinks to zero..... Ps I am just a highschool student so plz explain it in detail.....I have seen your multivariable and vector calculus playlist but I have severe doubt in understanding green theorem Thank you very much sir
I have a doubt here. So to get rectangular, we cut the shaded area into rectangles (which is require a lot or i can say infinetely cut). So we can't ignore the narrow boundary, can we?
Hello Dr. Bazett, Thank you for your thorough and easy to follow explanation! I still can't quite understand why the circulation density of a uniform rotation vector field is non-zero and the circulation density of the "whirlpool effect" is 0. The latter field is F=(-y/(x^2+y^2))i+(x/(x^2+y^2))j. Do you have any thoughts about this peculiar field? Thank you again for your videos!
practice practice practice. When you find something you are weak at, take note of the specific challenge. Then master it so you never struggle with that specific thing again. Math is often a lot of rather small details all put together so master all those details.
@@AshishSingh-753 So you're saying that if you got the equations without the words, you're fine? Write a "Givens:" and a "Find:" try to convert it into NOT a word problem, and then you don't have the word problem issue. Try easy word problems until they get hard as well and you're good.
Start with a generalized rigid body, spinning at a rate of ω, centered at the origin, spinning CCW around the z-axis At any given point on the body, its linear velocity is given by: v = Take the curl of this vector field. curl v = d/dx (ω*x) - d/dx (-ω*y) Carry out derivatives: d/dx (ω*x) = +ω d/dx (-ω*y) = -ω Thus: curl v = ω - (-ω) curl v = +2*ω
In the 2D case, curl is a scalar, that is positive when CCW and negative when CW. Since the curl is a scalar, imagine it as the height of a hill above a reference level we call zero elevation. The volume of this hill, tells us the total line integral around the vector field, enclosing that region, according to Green's theorem. Volume above the reference level we call positive, and volume below the reference level, we call negative. By convention of a right-handed coordinate system, we consider CCW rotation to be positive curl, and CW rotation to be negative curl.
@@DrTrefor We have weekly quizzes and your topics match with them. In fac this topic will be asked in tomorrow's quiz :P. Btw these are gr8 videos. Thank you a lot :D
