Given a surface

S

with implicit equation

F(x,y,z)=c

(constant) and given any\ curve

C

with parametric equation

r(t)=(x(t),y(t),z(t))

lying on the surface

S

,\ show that the gradient

gradF

at a point

P_(0)=(x_(0),y_(0),z_(0))inC

is orthogonal to the\ curve

C

. Deduce that

gradF(x_(0),y_(0),z_(0))

is normal to the tangent plane to the surface\

S

at

P_(0)

. What is the equation of that tangent plane?

Given a surface S with implicit equation F(x,y,z)=c (constant) and given any curve C with parametric equation r(t)=(x(t),y(t),z(t)) lying on the surface S, show that the gradient F at a point P0=(x0,y0,z0)C is orthogonal to the curve C. Deduce that F(x0,y0,z0) is normal to the tangent plane to the surface S at P0. What is the equation of that tangent plane