Consider the paraboloid S zx2 3y2 k 0 oriented with the ou

Consider the paraboloid S : z-x2 +3y2, (k > 0) oriented with the outward normal (a) Use a suitable parametrization x(u,v) and determine the unit normal vector N(u, v); dNp Determine if the point (0,0,0) is elliptic/hyperbolic/parabolic or (c) Find the Gaussian curvature K of the surface at a point p ES planar

Solution

. a) Steps to find normal vector:

If a vector at some point on S is perpendicular to S at that point, it is called a normal vector. If a normal vector has magnitude 1, it is called unit normal vector.

b)

Eigen vector of a matrix A is a vector represented by a matrix X such that when X is multiplied with matrix A, then the direction of the resultant matrix remains same as vector X.

Mathematically, above statement can be represented as:

AX = ?X

X represents the eigen vector.

c) Gaussian curvature or Gauss curvature ? of a surface at a point is the product of the principal curvatures, ?₁ and ?₂, at the given point. A sphere has gaussian curvatures everywhere and so in the given question it is said to be elliptic.