Let f be the function of two variables given by the formula

Let f be the function of two variables given by the formula x^2 + y^2 whose domain is the set of points in the plane satisfying x^2 + 2y^2 lessthanorequalto 1. The boundary of this function\'s domain is the plane curve C given by x^2 + 2y^2 = 1. Let D be the lift of this curve to the graph of f in 3-space and let F = be a vector field defined on all of 3-space. If S is the graph of f in 3-space, Find a parametrization R(s,t) of S in terms of parameters s and t and write down the constraint that these parameters must satisfy due to the restricted domain of f (which, recall, is not the whole plane).

Solution

let x = s and y = t . then f = s^2 + t ^2. hence in parametric form R(s, t) = (s, t,s^2 + t^2).

constrainst are simply that on x and y so s^2 + 2*t^2 <= 1