Question

What is a tangent vector ?
Suppose a graph w=(x^2) + 5 (y^2). What is the tangent vector to this graph ?

Answer 1

a *surface* , including this one, will have an unlimited number of tangent vectors at any point, all of which will lie in the *tangent plane* at that point. in the x and y direction, for the given example, these will be given by ∂w/∂x = 2x and ∂w/∂y = 10y. but these are only the directional vectors in the x and y directions.

it is the normal vector that will define the surface, or the tangent plane, at any point on the surface: the normal will be <2x, 10y, -1> for this example.