A Tangent Plane Problem

6760, Math 21a, Fall 2009

Week 5 Problems C, Math 21a, Multivariable Calculus

Course head: Oliver Knill

Office: SciCtr 434

Email: knill@math.harvard.edu

Problem C: The surface S with parametrization

r(u,v) = (u,v+u²,cos(u+v))

is shown to the left.

a) (4 points) Find an implicit equation g(x,y,z)=0 for this surface, the linearization L(x,y,z) of g at (1,0,1) and the equation L(x,y,z) = g(1,0,1) of the level surface of the linearization of L. This will be identified later as the tangent plane at (1,0,1).
b) (2 points) Why are the vectors r_u(u,v) and r_v(u,v) tangent to S?
c) (4 points) Use b) to find the equation of the tangent plane at the point (1,0,1) again.