I totally misunderstood this when I took Calc 4. Students don't realize how good they have it with MathTubers like yourself! I also walked up hill both ways to grade school in the snow... lol
I first learned vector calculus close to 50 years ago. I have listened to the first ten lectures this morning. They are an absolute joy. The lecture on hand mapping of vector fields brought back memories pre laptops of Schey's Div, Grad,Curl And All That. I hope in the future you will have videos on differential geometry. Your teaching method would be ideal. Thank you Leeber
7:20 I really admire and appreciate your teaching style. The way you used different colours to distinguish distinct terms from one another have actually won my heart. Actually this video needs more visualization i.e the scalar function M with i component is integrated along dx and much more. Anyway you did your best. Actually our teacher of English and philosophy used this trick of different colors to distinguish distinct points for more clever visualization and nice comprehending and I really wanted another teacher like him for 2 years and today I found another great teacher, Dr. Trefor Bazett. 💚💚💚💚💚😘😘😘 now I am anxiously waiting for your videos on green's theorem and much more.
Thanq Mr. Bazett, i enjoy and learn a lot through your vids. I would like you provide a feedback and hope u find this constructive: Could you attach few extra sound foam in the studio as I am hearing an echo/sharp sound while at full volume in my audio.
Huge thanks from South Korea! I' ve been looking for video that exactly covering this theme. one question sir, if i want to get line integral w.r.t x, ∫f(x,y)dx == ∫ Mdx is this right? and can this concept applied to get work in physics?
Do you know if there's any way to verify the line integral value *geometrically* when your projection has overlaps, as you mentioned about your projection onto the yz-plane? I am struggling to compute the correct value geometrically. Without overlaps, it works beautifully!
Ohh thanks a lot man, what i used to do while solving the example integral is that substituting y in x/y as( √y)/y =y^(-1/2) with limits of y varying from 1 to 4 (dy ) interestingly the answer remain same in this case . Can you pls clarify about the reason ? Is this approach is right ? If not , why ?
How to explicitly solve the integral ? What are possible condition to make any function explicit ? Is it possible ? please provide link if you have any video about it ? Or is it down the line in the playlist. Thanks for reply.
By explicitly solving the integral, what is meant is providing a method where you relate all of the related variables with functions. Basically, in your case, replacing x with sqrt(y) is explicitly defining x in terms of y. The reason you generally avoid this is that often this is hard to do, and you get problems with signs. For example, if the integral had been extended to start before 0, for x < 0, x=-sqrt(y). In this case, you would have to break the integral into parts, or parameterise.
The condition for being able to explicitly relate one variable to another is that the mapping to the variable you’re trying to describe in terms of the other, from the other, must be a function. That is, in this case, since there is only one possible x-value per y-value, it works. Otherwise, it doesn’t. If, for example, we examined across (-1,1) to (2,4), we cannot determine all of the x-values across the interval (-1,2) from y-values alone since x=+/-sqrt(y).
I have a question is this formula true for line integrals of any possible scalar function f(x,y) Let X = integral of f(x,y) with respect to x Y = integral of f(x,y) with respect to y S = integral of f(x,y)ds As ds² = dx² + dy² So is the formula below true? S² = X² + Y² or S = root of ( X² + Y² ) In short I mean to say that can we apply Pythagorean theorem here that the line integral of f(x,y) w.r.t ds is equal to the root of sum of squares of line integrals of f(x,y) w.r.t dx and dy? Why this formula is not written in any of the vector calculus books when line integral w.r.t only dx or dy is introduced ?
Sadly not, I believe. I think it is an issue with signs. The differential in the main line integral is intrinsically positive, we write ds = sqrt(g'(t)^2+h'(t)^2)dt. The idea here is we are always increasing the length of the curve as we go along. However, just dx=g'(t)dt can be positive or negative. And this is something going on locally in each little segment so the line integral with respect to x, say, could have a lot of internal cancellation along its path that the main line integral just doesn't see. SImilarly, in this video we saw an interpretation of the line integral with respect to x as a projection and then the area of that projection, but that ONLY works, when there are no overlaps etc which is quite a special case
I have this "condition", that I don't believe when people tell me something about a math topic. I don't believe in the books (Stewart's and the like). I've been lied to for many years. Now, I just don’t believe them anymore. My recent disbelief is about parametric equations. I feel as if everybody is lying about parametric equations, that it’s the work of a very small group of mathematicians and not standard at all. 🙄
At 7:52 you forgot to put the primes!
Ooooh thanks great catch 👍
Hey brother help me out with this topic..
❤
I totally misunderstood this when I took Calc 4. Students don't realize how good they have it with MathTubers like yourself! I also walked up hill both ways to grade school in the snow... lol
I'm glad that I live in this day and age where I can resort to people like you when I'm stuck with a bad teacher, so thank you
Dr. Trefor Bazett
I first learned vector calculus close to 50 years ago. I have listened to the first ten lectures this morning. They are an absolute joy. The lecture on hand mapping of vector fields brought back memories pre laptops of Schey's Div, Grad,Curl And All That. I hope in the future you will have videos on differential geometry. Your teaching method would be ideal. Thank you Leeber
SO this is where Mdx and Ndy come from!!!!
You sir opened my mind.This all makes so much sense now.
glad it helped!
7:20 I really admire and appreciate your teaching style. The way you used different colours to distinguish distinct terms from one another have actually won my heart. Actually this video needs more visualization i.e the scalar function M with i component is integrated along dx and much more. Anyway you did your best. Actually our teacher of English and philosophy used this trick of different colors to distinguish distinct points for more clever visualization and nice comprehending and I really wanted another teacher like him for 2 years and today I found another great teacher, Dr. Trefor Bazett. 💚💚💚💚💚😘😘😘 now I am anxiously waiting for your videos on green's theorem and much more.
Please make a playlist for differential geometry/bless us with a nice derivation of the generalized stokes theorem! You are amazing at presenting math
Thanks a lot Sir . I started today and i am at video number 10 . Love you
Great job! Looking forward to seeing you at the end:)
Very intuitive explanations!! Thanks alot :)
Glad you enjoyed!
Your videos are the best! You are a math life saver.❤
Thanq Mr. Bazett, i enjoy and learn a lot through your vids. I would like you provide a feedback and hope u find this constructive: Could you attach few extra sound foam in the studio as I am hearing an echo/sharp sound while at full volume in my audio.
Thank you very much for your videos!
Huge thanks from South Korea! I' ve been looking for video that exactly covering this theme.
one question sir, if i want to get line integral w.r.t x,
∫f(x,y)dx == ∫ Mdx
is this right?
and can this concept applied to get work in physics?
thanks,
Why other videos in this plalist don't open ? e.g video about Flux ,I can't play it
They're all coming out (2 a week right now), but I upload all videos about a week early for members of the channel before making the fully public.
Thank you .i wish to you happy with you family all of entie life
Thank you very much 🔥🔥🔥
Do you know if there's any way to verify the line integral value *geometrically* when your projection has overlaps, as you mentioned about your projection onto the yz-plane? I am struggling to compute the correct value geometrically. Without overlaps, it works beautifully!
Great video, thanks :)
Thanks alot for this video . 🙏👌
You're most welcome!
THANKS
Ohh thanks a lot man, what i used to do while solving the example integral is that substituting y in x/y as( √y)/y =y^(-1/2) with limits of y varying from 1 to 4 (dy ) interestingly the answer remain same in this case . Can you pls clarify about the reason ? Is this approach is right ? If not , why ?
Provided you can explicitly solve one variable for the other, which is rare, then yes this works
How to explicitly solve the integral ? What are possible condition to make any function explicit ? Is it possible ? please provide link if you have any video about it ? Or is it down the line in the playlist. Thanks for reply.
By explicitly solving the integral, what is meant is providing a method where you relate all of the related variables with functions. Basically, in your case, replacing x with sqrt(y) is explicitly defining x in terms of y. The reason you generally avoid this is that often this is hard to do, and you get problems with signs. For example, if the integral had been extended to start before 0, for x < 0, x=-sqrt(y). In this case, you would have to break the integral into parts, or parameterise.
The condition for being able to explicitly relate one variable to another is that the mapping to the variable you’re trying to describe in terms of the other, from the other, must be a function. That is, in this case, since there is only one possible x-value per y-value, it works. Otherwise, it doesn’t. If, for example, we examined across (-1,1) to (2,4), we cannot determine all of the x-values across the interval (-1,2) from y-values alone since x=+/-sqrt(y).
Clear
There is a synchronization problem between the audio and visual. Distracting
Just a question ?.at 10:50 the answer we got is 2 so this is actually the area of the surface formed between the parabolic curve and the function ?
NIIIIIIIIIIIICEEEEEEEEEEE
I have a question is this formula true for line integrals of any possible scalar function f(x,y)
Let
X = integral of f(x,y) with respect to x
Y = integral of f(x,y) with respect to y
S = integral of f(x,y)ds
As ds² = dx² + dy²
So is the formula below true?
S² = X² + Y²
or S = root of ( X² + Y² )
In short I mean to say that can we apply Pythagorean theorem here that the line integral of f(x,y) w.r.t ds is equal to the root of sum of squares of line integrals of f(x,y) w.r.t dx and dy?
Why this formula is not written in any of the vector calculus books when line integral w.r.t only dx or dy is introduced ?
This was the question I wanted to ask you since September. 😅
Sadly not, I believe. I think it is an issue with signs. The differential in the main line integral is intrinsically positive, we write ds = sqrt(g'(t)^2+h'(t)^2)dt. The idea here is we are always increasing the length of the curve as we go along. However, just dx=g'(t)dt can be positive or negative. And this is something going on locally in each little segment so the line integral with respect to x, say, could have a lot of internal cancellation along its path that the main line integral just doesn't see. SImilarly, in this video we saw an interpretation of the line integral with respect to x as a projection and then the area of that projection, but that ONLY works, when there are no overlaps etc which is quite a special case
@@DrTrefor Oh I got it now . Like in this video the y overlaps were cancelling each other out. Thanks sir.
Thank youu❤❤
Very helpful, thanks. I believe that there are significantly more left handers who are gifted mathematicians. A set to which you belong.
谢谢！
at 7:40 the M(x,y) need to be M(g(t), h(t)) why did you say the equal M(x,y)?
I think I'm the only one to notice that
I guess itʼs because thatʼs what g(t) and h(t) return.
👋 Good
🙏🙏🙏
note: reached here 26/2 not yet watched
👍👍👍👍👍👍👍
dale pau
I have this "condition", that I don't believe when people tell me something about a math topic. I don't believe in the books (Stewart's and the like). I've been lied to for many years. Now, I just don’t believe them anymore. My recent disbelief is about parametric equations. I feel as if everybody is lying about parametric equations, that it’s the work of a very small group of mathematicians and not standard at all. 🙄
This is pretty cool, but have you heard of battletoads?
Thank you very much for your videos!