Projection stereographic and second fundamental form

Simone Furcas · May 9, 2015

Let r:R2 →R3 be given by the formula

$rac{2u}{u^2%20+v^2%20+1},%20\frac{2v}{u^2%20+v^2%20+1},%20\frac{-1+u^2%20+v^2}{u^2%20+v^2%20+1}).gif$

Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n).

I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained impossibile result... The solve is too long to write down here... I use I used

$gif.latex?N=\frac{ruXrv}{||ruXrv||}.gif$

micromass · May 10, 2015

Think geometrically: what vectors are orthogonal to the surface of the sphere?

Simone Furcas · May 10, 2015

I was thinking it was very difficult... Now I think to be a bit silly! :) thx

Simone Furcas · May 12, 2015

Is it the same with an ellipsoid (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ? N is (a*cos(x)*sen(y),b*sen(x)*sen(y),c*cos(y)) ?

micromass · May 12, 2015

Projection stereographic and second fundamental form

Graduate Nonautonomous Lie derivative

Graduate A nice YouTube channel about all things Riemann Zeta

Graduate Equivalent definitions of tensor field

Undergrad What is "x" in this definition of a basis?

Graduate Knots In 4D

Graduate Nonautonomous Lie derivative

Graduate On covariance of the Lagrange equations

Insights Revisiting the Velocity-Time Function

Insights Remote Operated Gate Control System

Insights AI Enriched Problem Solving

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Projection stereographic and second fundamental form

Similar threads