Computational Proofs for Tangent Vectors and Differential Geometry of Surfaces

[chaotixmonjuish]

Homework Statement

Given the following:
Some surface M: z=f(x,y) where f(0,0)=f_x(0,0)=f_y(0,0)

and
U =-f₁U₁-f_yU₂+U₃}/Sqrt[1+f_x²+f_y²]
and
u₁ = U₁(0)
u₂ = U₂(0)
are vectors tangent to M at the origin 0.

We want to prove
S(u₁)=f_xxu₁+f_xyu₂

My problem here is conceptually wadding through this. I really just need a computational push.

 

[lippyka]

Homework Equations f1=partial derivative of f with respect to xfy=partial derivative of f with respect to yfx2+fy2=magnitude of the gradient at (x,y)The Attempt at a SolutionI'm trying to do this by taking the partial derivative of U with respect to u1 and u2. So far I have:dU/du1 = -f1-fy(dx/du1)+(dx/du1)/sqrt[1+fx2+fy2]dU/du2 = -fy-f1(dx/du2)+(dx/du2)/sqrt[1+fx2+fy2]I'm not sure what to do next.