Dr Peyam, Thank you for explaining very clearly (and enthusiastically as always) the normal derivative. Ideas like this one are often either assumed to be too easy to be discussed in detail or just mentioned during some elaborate calculation. I do not know if you have a video on Lipschitz domain which is also about the boundary of a domain.
If u is a function of |x|, the normal derivative of u on a surface of constant |x| is obviously going to be u' of |x|, according to the explanation of the normal derivative. The example is really checking that the definition gives the right result for a case where we know what the answer should be, so it's good that we aren't surprised.
Hi Dr. Peyam!!! I haven't stopped in for quite some time but wanted to for this video. In Calc3; partials, gradient, and the TNB space was one of the biggest "Ah HA!!!!" moments in my maths career. It was also one of the most exciting because of all the immense power you now wielded. TNB was very hard for me to understand initially but once I realized it was another Cartesian coordinate plane "flying" around in the primary coordinate plane, anchored to the curve in question. I envisioned it as a roller coaster track and I am in the coaster car riding along, and me sitting in the car is one frame of reference, the TNB frame. Riding through the primary space of x,y,z. Once I realized that, it was like an avalanche of realizations all happening in short order. It was amazing TBH. So yeah, these topics are really awesome. I am in diffeq now and due to being "taught" remotely, I am failing horribly. Its not even failing, its like I understand absolutely zero. I will be retaking it in the spring, assuming we are back to REAL school and not this remote learning "socio-political experiment" or whatever you want to call it.
What is TNB? Tangent, Normal, and Binormal vectors, I am guessing? Just never heard it referred to like that before. Makes sense though, their being orthogonal and all!
Dr Peyam I don’t really understand the difference is there an equation that relates the two functions using the gradient or does curl have anything to do with it?
There’s no equation relating them since they’re two different things. The only relationship maybe is that if you integrate the divergence of the gradient, by the divergence theorem you get the gradient dotted with the normal vector which is the normal derivative
@@drpeyam The curve have the representation (x,f(x)) at each points in it, so can we think about (1,f'(x)) as the gradient w.r.t x of the u where u= (x,f(x))?
