Before I met this channel, I thought the best explanation for this concept is MIT OpenCourse Ware Multivariable Calculus. After I watched this, this is even better. Fantastic course and explanation and handsome face haha.
I completed all the videos in this playlist, and I've also seen a lot of videos of yours. Only thing I can say, YOU'RE GREAT. Take a bow. Heartiest gratitude from Bangladesh 🇧🇩
Prof, I have been following your videos for some times now and I can say that you bring The Big Picture into mathematics through your very clear explanations, academically correct yet without high-sounding jargons (i.e. technical terminologies). Merci pour vos bons œuvres.
Thanks Dr Trefor, you are a special teacher, who explains things precisely, and actually understands your subject matter expertise… I now understand Greens Theorem perfectly! If math exams were done verbally, and based on how well a lay person understood any math topic… you would score 100%… ( as long as you had the visualization technology)
I love the videos when it is about the tiny world of mathematics believing that if one understands how it works on the smallest scale one will also understand the macroscopic effects of it.
Isn't that so cool?! Being able to translate local to global information seems too good to be true, but it works out! (And thanks for becoming a member, much appreciated:))
It's also cool that this is true for physics as well: quantum mechanics (microscopic) is a generalization of classical physics (macroscopic). It's called the Correspondence Principle if you're interested in learning more about it. The symmetry between mathematics and reality is really mind-blowing.
you're the reason I'm passing multivariable calc. For some reason vector calc is a part of that course as well lol. Rutgers is no joke when it comes to math courses...
Thank You So Much, Dr!!! I still have Stoke's and Divergence Theorem and your course really helped me understand the background and the actual meaning of these chapters in all math courses this academic year which was difficult for me to cope with and with the pandemic! I would also like to ask you if you are going to upload a Probability and Statistics course to the page because I am taking it next semester (Fall 2021), because they will help me a lot and will make the course more interesting to me. I am so pleased to have found your youtube channel which was very helpful to me.
Trefor, I tried to derive the flux formula around the small rectangle, but my derivation is a little off. I'm getting extra factors in front of my dM/dx and dN/dy: (1/|delta y| dM/dx + 1/|delta x| dN/dy) (delta x)(delta y). Are we still assuming that n is an outward unit normal vector? Are we assuming that the small changes in x and y (delta x and delta y) are both one? I appreciate any help you can give. Thanks so much!
Hmm. Yes, outward normals, and no delta x and delta y are arbitrary small things, not necessarily 1. I'm guessing your issue involves maybe missing the denominator of the difference quotient in the definition of the partial derivatives. We have things like( M(x+delta x)-M(x))times delta y. We replace that with (dM/dx times delta) times delta y
@@DrTrefor - Okay, I am missing something. (M(x+delta x)-M(x)) comes from the right and left sides of the rectangle, correct? On the right side of the rectangle, isn't the outward normal pointed towards the right in the i direction? So wouldn't the unit outward normal be (delta x)/|delta x| in the i direction? I appreciate your response whenever you get a chance!
@@DrTrefor I figured it out. I was evaluating F at the wrong corners. For the top, I was evaluating F at the top right corner, and for the left side, I was evaluating F at the top left corner. Thanks!
I've a quick question. When we take divergence of every infinitesimal point and is essentially cancels out, does it have any contribution to the final answer?
7:23 If everything in the middle cancels out, is the net effect of the integral (over the region) the sum of the "tiny rectangles" along the border, since everything else cancels out?
Could the divergence formula be derived from the fact del dot F = delM/delx + delN/delx? The dot product sort of represents the magnitude of change for the vector F(x,y) and this then represents the "outgoingness" of F at (x,y), which you can then integrate over the region R to give you the "outgoingness" of R?
Hi, Dr. Trefor. I think there is an error in the first line of your description. You write, "In the previous video in our Vector Calculus Playlist (ru-vid.com?list...) we saw Part I of Green's Theorem, which related the local property of *Flux* (aka circulation density)...." Do you mean "the local property of *Curl* (aka circulation density)"? Thanks.
This might be a stupid question re.: 3'40" In going around the rectangle the N vector flips sign between the left and right (i vector) bottom and top (j vector) and in so doing produces a sum rather than a difference. For example along the bottom we get n=-j so F.n = -N; on the top n=+j but the the delta X is negative so there is a negative there as well. (n=normal; N is the y component of F). Where am I going off track?
You are most likely forgetting to remove the sign of the displacements (delta_x & delta_y). In the derivation of the circulation-curl form of Green's theorem we are concerned w/ F.dr where dr has both magnitude (delta_x or delta_y) and direction (giving -delta_x or -delta_y). But for this scenario, we are concerned w/ F.n ds, where n is the normal vector like you mentioned w/ direction -1 or 1 in x or y but the displacements (ds) should only be considered in their magnitudes (delta_x or delta_y w/o signs). This should give you the final relation shown in the video.
i used this formula to get the outward normal N=dy(i)-dx(j) which is derived from rotating 90 degrees the tangent vector T, which is given by T=dx(i)+dy(j). Now at the top (tangent)T=-dx(i) from the normal formula we replace dx(in Normal formula) by -dx. therefore get N=-(-dx)(j) which equals to N=dx(j). Or one more way is, magnitude of the normal here is going to be same as that of Tangent(they both are unit vectors, the magnitude comes from ds(change of arclength)), so only direction is to be determined which is, at the top, +j and the magnitude here is dx. Therefore N=dx(j).
Dr. Bazett, a great video, thanks a lot! I wanted to ask you a question about the Green's Theorem for Flux: I believe the Flux version of Green's theorem is also called the Divergence Theorem, or as İ leaned it, Gauss's theorem. The way I learned Gauss's theorem is a more generalized version of what you present in this video: Instead of C smooth, simple, closed curve, we have C d-dimensional compactum with C^1-Boundary. Now my question is, is there actually a difference between what you present in the video above and Gauss's theorem other than that Gauss's theorem is a generalization to d dimensions?
Nice catch, thank you! I'm actually really loving having the early member access because you guys get to be my proof-readers and I can fix it before it goes live to everyone:D
Never mind, I seem to have figured it out now. Though I'm a little confused. Solving it required me to apply limits (such delta y going to 0 and delta x going to 0) in order to make the expressions equivalent to the numerator form of the partial derivatives. Why doesn't this apply to the whole equation? For example : (M(x + dx, y) - M(x, y + dy))dy became (M(x + dx, y) - M(x, y))dy but I can't tell why I'm allowed to let the dy inside the brackets equal 0 without doing the same to the one outside of the brackets
All capacitors are flux capacitors, since electric flux (this concept applied to electric fields) is a concept used for deriving the capacitance in Farads.
Since d has another full time job in calculus, it's tradition to use the next consonant (s) to represent distance and displacement. ds is the infinitesimal displacement along the path.
I completed all the videos in this playlist, and I've also seen a lot of videos of yours. Only thing I can say, YOU'RE GREAT. Take a bow. Heartiest gratitude, from Bangladesh 🇧🇩
