Changing the Frame of Reference to study a Surface Eves Exercise 14.3-5

In the paper Changing the Reference Frame, we solve exercise 14.3-5 from Howard Eves' "A Survey of Geometry, Vol. 2" in a straightforward argument involving the gradient vector as the normal to a surface. A change of coordinates (frame of reference) allows the Taylor expansion at a point to be approximated by the second degree terms of that expansion. The main purpose of the paper is to study an example which showcases the algebraic tools needed to affect a change of coordinate axes in Euclidean space. The key insight is that a translation or rotation of the axes is equivalent to the inverse translation or rotation of the surface point set in the original reference frame. Although the example is a quadratic itself, f(x,y,z) = z + x^2 + y^2 = 0, for illustrative purposes, the result applies to any surface which can be written in the form z - g(x,y)=0 in the new reference frame.