Mathematics lectures and examples. I am an Associate Teaching Professor of Mathematics. On this channel are videos for Multivariable Calculus, Single Variables Calculus II, Mathematical Modeling (intended primarily for high school math teachers), and others. I am currently recording some lectures on Differential Equations, and hope next to make a series on proof writing and Real Analysis.
I am pretty skilled with MATLAB--if you are wondering how I made a graphic or animation, it was probably in MATLAB. (I will eventually put some MATLAB videos up here too...)
You're welcome, and thank you for this kind remark! I was motivated to make this video because I felt a lot of presenters were overcomplicating the geometry (throwing circles on the ground, etc). If there are any similar "recreational" topics that you would like to see, please let me know.
I had a question about Ex 2: If the hemisphere were a hollow surface as opposed to a solid (in the example), then does the parameterization of the surface reduce down to that of a curve? r(v) = < 2cosv, 2sinv, 0> . Thanks.
I paused the video at 18:39, 18:41/18:42, the TxN direction seems to be opposite to that of B by right hand rule, seems to match NxT. Is that some viewing/ right hand error rule from my end?
Gosh I agree, it really does look that way! I don't have my matlab code handy to check for an error but I think it might be an optical illusion (this happens with 3D plots). If you pause right at 18:27, you can convince yourself of two different configurations. If you imagine N as point back into the screen, then TxN is opposite of B. But if you imagine N as pointing out towards you, then TxN is indeed B (this is the case). Around the 18:40 mark, I think what's happening is that the curve is actually bending differently than it looks in the 2D flattening picture. Right at 18:40, my interpretation (and yours, I think) is that N is pointing "out of the screen". Instead, I think the curve is actually bending away from you at that moment and N is pointing into the screen. (Or... my code might have an error that comes out at that peak a few seconds earlier.)
@@bevinmaultsby 18:27 - I get the two configurations. But why should N point outward at all opposite to the direction of bend? 18:40 - to be honest I just picked out T and N at that point and tried the right hand rule. Maybe if that portion of the curve was directly facing me as opposed to being on the side the orientations would be clearer i.e. as you had mentioned optical illusion. Looking at the circle portion of the curve, the orientation of B is as expected. So, by Occam's razor this is an optical illusion problem at the points where the direction of B seems to be off. Edit: 18:27, if I turn my head around a bit so that I directly face the curve, the direction of B is rightly as expected.
Yeah, it sounds like you've reconciled yourself to my confusing graphic, but I meant that the bend is opposite of what it seems (so N is pointing "inward," but inward looks outward).
@@bevinmaultsby Thanks, I suppose it is the nature of the beast that one ought to change the orientation of the graph, move it in a way so that things make sense. For now, it is okay for me to proceed ahead as I am learning a lot. 👍
@@bevinmaultsby You are an excellent teacher, so any fault certainly does not lie with you. I think a lot of the problem stems from the fact that most people seem to view education as a punishment, or simply as a means to an end that must be suffered through to reach their goal...a necessary evil if you will. I have always enjoyed learning, but came to math relatively late, by necessity because I needed to be able to understand math and physics for electronic engineering, which than branched into specializing in magnetics. I soon discovered that I love math and physics simply for their own sake, and now there is no turning back!
I tutor math all the time and honestly hated that I didn't know how to derive this identity. Thanks so much, knowing where formulas come from is so much better than just knowing the formula!
Excellent! I think there is a more visual/geometry based way but it's much trickier and not something I've ever actually worked through. Euler's formula is a great tool for trig identities.
Thank you for all the videos, Dr Maultsby. I have a Q in this video, though. In the Matlab demo one can see that the unit tangent vectors and tangent vectors, are both tangent to the curve at various t. Why is the dot product of the unit tangent vector with r(t) not zero? Thanks.
@@bevinmaultsby No worries. I suppose for some weird reason, I was equating the word tangent (or derivative) to that used in a context of a circle : the tangent being orthogonal to the radius and hence their dot product be zero. For r(t) = (t,t,t), tangent vector r'(t) = <1,1,1> This would point in same direction as r(t), dot product is not zero. Is this "my brain has frozen up" moment?
Sounds like you connected the dots! There's no special relationship between the position vectors r(t) for a parametrized curve and the tangent velocity vectors r'(t). You are right that when we consider a parametrization of the circle, r(t) . r'(t) is 0, but that's a special case. [You may have also been thinking of how r'(t) . r''(t) = 0 for any constant-speed curve (but not in general).]
@@bevinmaultsby Thanks for replying, I appreciate your time. You had mentioned "unit speed" a couple of times and then "constant speed" as a prereq for r'(t) and r''(t) to be orthogonal, along with the proof as well, so that was not the issue. The circle being a special case tripped me up... I have been introduced to Calc III earlier but it was mainly a plug-and-chug scenario - a waste of time. OTOH, you motivate the topic and its development which leads to better appreciation of the topic. I am not a Math major/ aspiring grad student in but I am curious if you teach any course on proofs that act as a leg up for advanced courses? I see some Real analysis II videos on your channel and would like to go through those, but want to start by going through a basic proofs course to build a math temper/mode of thinking to begin to solve problems in Spivak and Apostol and then widen the net. Thanks again.
I am teaching Real Analysis II this semester, hence the uploads (and the pace of the uploads, which I'm actually bit behind). However, the next thing in the pipeline, so probably around Nov-Feb, is proof-writing and Real Analysis I.
I hope this was useful! I do not do disease modeling, I just think this example is interesting to learn while studying differential equations. You may want to look at variants, like the SIRV model (which includes vaccinations).
It's a lightboard. I write on clear glass facing the camera, then the image is flipped and text and pictures are added. It's pretty fun but took practice.
Thank you for this video. Currently studying control theory and this video clarified some things about Laplace transform. I also found handouts from MIT 2.14 Analysis and Design of Feedback Control Systems on Understanding Poles and Zeros helpful. Liked the video.
very useful for my exam tomorrow. I am a pre-med but recently picked up a math minor and your videos have been useful to scratch up on since I have not taken a single math class except multivariable calculus as a dual enrollment since 12th grade.
@@bevinmaultsby You're welcome and Don't be hard on yourself. I have been watching such scientific videos a lot. You have one of the best. Your handwriting has the features of a woman's elegance.
Oh great! I'm teaching in this semester and will be adding every week. It's a pity not to add Analysis I first, but I will circle back to that once Analysis II is completed.
Yes, we can essentially think of it as computing a 4-dimensional "hypervolume." From the point of view of the Riemann sum, we chop up the domain into cubes, and to each little cube, we associate a function "height" value f(x*,y*,z*). Then f(x*,y*,z*) ΔV (or ΔxΔyΔz) is like height * volume of the base to get a 4-dim volume.
@@bevinmaultsby the confusing part for me is with the differentials, isn't the parametric surface a 2D object? corresponds to 2 differentials du and dv, which gives us dS, which should be a "tiny change in area" (still in R2) , or you mean we need to think of du dv in terms of the pullback f ⚬ r ? which i guess is a small change in R3?
New in 2024: you first need to flip on Cengage as an LTI External Tool. See the image at the top of my page here maultsby.wordpress.ncsu.edu/resources/
What a great visual explanation! I came here as the theorem is used in helping evaluate the Gaussian integral, but wanted to understand why the theorem itself is valid. Your explanation was easily understandable even with minimal knowledge of multivariable calculus. Thank you very much for putting this video out :)
Oh, it looks like I changed my mind about what I wanted to do on the first line and didn't update the second line. That second sentence should be "Then find an equation for the normal line to the graph of x^2 + y^2 + z = 9 at (1,2,4)."
