6760, Math 21a, Fall 2009
Week 5 Problems C, Math 21a, Multivariable Calculus
A Tangent Plane Problem
Course head: Oliver Knill
Office: SciCtr 434
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 ru(u,v) and rv(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.

Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University