Orientable vs Non-Orientable Surfaces and the Mobius Strip

One property that a surface may or may not have is orientability. Loosely, this means it has two distinct sides. For example the surface of a sphere has an inside and an outside, but a Mobius Band only has one side. More specifically, a surface is orientable if we can continuously assign a field of normal vectors, which is like choosing one of the two sides. This is going to be useful for us in Vector Calculus as we will be talking about the flux across a surface from one side to the other, and so we will want to restrict to be talking about orientable surfaces.
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[@DrTrefor]

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[@DrTrefor]

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[@issolomissolom3589]

Excellent explanation Thanks 👍😌

[@ManojKumar-cj7oj]

This mobius strip is blowing my mind🤔

[@raviphanindra9454]

thanks a lot for your clear explanation.

[@zareshkhan2294]

nice and good visualization

[@DrTrefor]

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[@nehalkalita1]

Nice explanation on mobius strip.

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thanks! useful and clear;

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[@DrTrefor]

Glad to hear that!

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the person who disliked failed this course

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Blows my mind a bit,.

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Nice video :)

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[@DrTrefor]

Thank you so much!

[@korky-7349]

Signed, Sincerely yours_ (From a person this been reading much on the fascinating ,ongoings at this place and its numerous Physical science affiliated implications!) Korky= Carl Aserio Yonkers NY .

[@chinmoyboruah9549]

Thats awesome

[@DrTrefor]

Right?!?!?

[@JoseRamirez-pt7rn]

Hiiiii. Hope you are well. I wanted to know, could you help me find cases of non-orientable surfaces other than morbius tape?

[@tahamr83]

Thank you such amazing content Dr. Trefort Bazett. What tools you use in your videos for your 3D graphs? Any other tools you recommend? Preferably that are free and open source. Thank you once again.

[@DrTrefor]

I mostly use MATLAB but I think geogebra is best for students (and free)

[@kimbalcalkins6903]

Would a polarized mobius strip change polarization when traversed ?

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[@chandrakantpandey1925]

Sir, In R2, we have two different types of line integrals, the tangential line integral ∫F⋅dr and the normal line integral ∫F⋅nds. However, for all the surface integral, I have only seen the normal surface integral defined by ∬F⋅ndS is there a similar concept that is close to the tangential line integral ∫F⋅dr?

[@chandrakantpandey1925]

To give a motivation, these two different integrals are very helpful for understanding Divergence theorem and Stoke's theorem. Let F=Pi+QjF=Pi+Qj. The Divergence Theorem says ∬div(F)dxdy=∫F⋅nds. This gives ∬(Px+Qy)dxdy=∫F⋅(−dy,dx)=∫Pdy−Qdx Stoke's theorem is ∬curl(F)⋅kdxdy=∫F⋅dr and this gives ∬(Py−Qx)dxdy=∫F⋅(dx,dy)=∫Pdx+Qdy. Green's theorem can be derived from either of the two above theorems.

[@DrTrefor]

Oh this is a great question and I didn't explicitly address it in the series. In short, no there isn't a great analogy of circulation/flow from the curve to a surface. Basically the issue is that on a curve there is one direction and we can talk about how the field aligns with that one direction. But in a surface there is too much degrees of freedom to make sense of an analogous concept. We do have stokes theorem that looks at the curl at each point along the surface.

[@chandrakantpandey1925]

@@DrTrefor Stokes theorem looks for the normal component of curl along each point of the surface . Yes you're true there would be no meaning of tangential component of curl across any surface . Thanks a lot sir 🙏. Your videos are really rocking ❤️ ! Please let me know if you're making videos on Real Analysis (sequences and series) in near future . It would be a great help for my semester exam in next Jan . Thanks again !!

[@kaiyin204]

but what if you rotate the mobius strip twice instead of once? you get 2 sides

[@Jewelsonn]

There are two sides obviously. The other side's dimensions are paper thin. That's why we cannot see the whole mobius strip. Why is this hard to grapse? Too distracted to see the lines drawn on one side?

[@placeholdername2222]

Here after the "Why did the chicken cross the mobius strip" joke

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[@ricardobautista-garcia8492]

Fourth comment. Take care.

[@DrTrefor]

haha fourth is pretty good!

[@ricardobautista-garcia8492]

@@DrTrefor True.

[@saadahmed688]

I liked the part where the strip was morbing

[@Wstarlights]

You literally just entered 2d.

[@korky-7349]

A mobius strip "appeared" to a scientist at the infamous Skinwalker Ranch, in a location in a room ,such that one and only one person if ~6 in the room could see it. ( the rocket scientist) This person was there to be briefed in the odd activities ocurring at that place. My question to you, as excellent mathematician that you are...is that, if this appearance of a Möbius was in intended from whomever or whatever , was either sending some type of message to this very intelligent , very credible scientist, or, engendering a subliminal response in the awareness of said scientist, What ,possibly is the message? What unique properties of the Möbius strip could be a symbol of (1) something the strange foreign intelligence wished to convey to the scientists there ...or, (2) Possibly could you attempt to speculate what about the Möbius , might be representative of the foreign intelligences presence at that place? I realize good sir that thus udxan impossible question . I am attempting to glean what about the Möbius renders but suitable to be symbolic of the physics of UFOs , gravitation, quantum physics, consciousness.. Please, if you please,take a swing at it ??/! Thank you! And, I am overjoyed to find your lecture , and absolutely shall be "attending" more of your rigorous,, clear ,fine mathematics instruction...

[@jokerjackass8461]

👍