Tangent plane to a surface

We now want to move on from two dimensions to three dimensions, and therefore we need to deal with surfaces in 3D. In this video, I illustrate how to parametrize a surface in three dimensions, and as an application, I show how to find the equation of the tangent plane of a surface at a point. Be prepared to see our good old normal vector again :) Enjoy!

[@tomatrix7525]

This series is just getting better and better! By the time I actually take vector calc next semester I’ll be an expert thanks to you

[@bandamkaromi]

Thanks, Dr. Peyam.

[@drpeyam]

My pleasure :)

[@aarondelacruzruiz3416]

Haha. Amazing. Muy bien.

[@loganreina2290]

Dr. Peyam, is it not the wedge product that is more important as it is more general, whereas the cross product is defined in three dimensions? I'm still early in my undergrad, so maybe I'm mistaken.

[@drpeyam]

You’re absolutely correct :)

[@srpenguinbr]

I was thinking about it two weeks ago

[@adeelali8417]

Isn't a curve 2 dimensional and a surface 3? A curve is on an x,y plane isn't it? Surface is on an x,y,z?

[@patricksalhany8787]

I always thought about what if for a surface, the normal vector at a point is 0, i.e., what if ru×rv=0, i.e., what if ru=lambda.rv (lambda=/=0), what will the surface at this point look like?
I asked my teacher when I was taking this course(vector analysis) and he seemed a little bit surprised, I don't know why, and told me that he doesn't know.
Dr. Peyam or anyone can help ?

[@drpeyam]

That's a good question, and someone asked me that too! Essentially what happens in that case is that the surface is degenerate at the point; think the analog of a cusp in single-variable calculus. It's the analog of finding the tangent line for f(x) = |x| at x = 0

[@patricksalhany8787]

Yes I thought about it, but I still have problems with visualising the surface at that point, because at that point N=0, which means that the normal vector to the given surface at that point is pointing to all the directions of the space, so do you think dr. that it will be pointy at that point like |x| at 0 ?

[@Koisheep]

There are several ways you can have ruxrv=0, and a whole theory behind. It's called singularity theory and it tries to classify all singularities in order to know whether or not you can force a tangent plane to exist at that point (now I'll say how).
For example, your surface around that point may look like the butt and tail of a swallow, in which case you have a swallowtail singularity (demonstrations.wolfram.com/TheSwallowtailSingularity/). Basically, a swallowtail singularity happens when you take the polynomial
x²+bx+c
and try to plot it as a function of b and c. In that case, you can define the tangent plane using something similar to taking limits over the derivative in 1D.
Nonetheless, there are other cases like the whitney umbrella (mathworld.wolfram.com/WhitneyUmbrella.html) in which there is no way to put a tangent plane (the bad points are where the surface cuts through itself)
Don't blame your teacher for not knowing about this; this is advanced differential topology theory, which is way beyond the standard math curriculum and totally unrelated to most mathematical analysis curricula (there's some overlapping with complex analysis tho, since it's esentially analysis over the sphere).

[@drpeyam]

^ That is amazing, thank you!!!

[@patricksalhany8787]

Thank you.

[@RiaziMohandesi]

Hi, do you Persian?

[@drpeyam]

Yeah

[@RiaziMohandesi]

@@drpeyam چه جالب، کجایی هستی دقیقا، تو ایران نیستی احتمالا که این ساعت بیداری؟

[@okokok4321]

Did you say спасибо at the end lol

[@drpeyam]

Yeah, hahaha 😂