The Most Remarkable Theorem: Part II (Surface Curvature)

Theorema Egregium Part II: Surface curvature.
#artofproblemsolving #gauss #calculus #algebra #calculus3 #vectorcalculus
#theoremaegregium #gaussiancurvature #curvature #extrinsicgeometry #intrinsicgeometry
#surfaces #curves #tangentvectors #normalvectors #movingframe #frenetserretframe
#parametrizedcurves #parametriccurves #parametricsurfaces #vectors #vector #mathematics
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[agrajyadav]

I have no right to be watching this for free. You are a great teacher, this is excellent!

[ToothbrushMan]

I think that your definition of a principle direction isn't quite there?
At any point on a surface, you can measure the line curvature of any line going through that point, the line being in the flat plane that is perpendicular to the surface at that point (i.e. its normal vector lies in the tangent plane at that point). You can then rotate that plane about the normal to the surface at the point and observe the line curvature vary as the plane is rotated.
In a 360 degree rotation, there will be two angles (180 degrees apart) where the line curvature will be a minimum. And there will be two angles where the line curvature is a maximum.
These directions define the principle directions, and the product of the maximum and minimum line curvature is the Gaussian curvature?
The principle directions cannot be the tangent vectors in the directions of the axes in whatever coordinate system the surface is being mapped in, as that would mean the Gaussian curvature would be dependent on the coordinate system being used, and wevknow that the Gaussian curvare is a property of the surface only, and not the coordinate system.
Please forgive me if i got any of this wrong.

[АлексейТучак-мч]

But what are principal directions?
If for example we rotate a hyperbolic paraboloid z=x^2-y^2 along OZ will they change? Or are they just sections of a surface by a surfaces of constant u or v?