Multivariable Calculus (Math 214) Assignment

due Wednesday, October 13, 1999

1. Find an equation for the plane that is given parametrically by

   (2, -5, 4) + u (2, -1, 2) + v (1, -2, -2) .

2. Find a parametric representation of the line in space given by the simultaneous linear equations

   2 x - y + 2 z	=	5
   x - 2 y - 2 z	=	1 .

3. Find a parametric representation P + u V + v W of the plane that is tangent at the point
   P = (-1, 1, 1) to the surface

   x^{2} y - z^{3} = 0 .

4. Let f be a function of two variables. The graph G_{f} of f is the surface in space defined by the equation

   f(x, y) - z = 0 .

   For fixed values of a and b let P be the point (a, b, f(a,b)).

   1. Find a vector that is normal to the surface G_{f} at the point P.

   2. Find an equation for the tangent plane to G_{f} at P.

   3. Find vectors U and V for which the expression

      P + u U + v V

      is a parametric representation of the tangent plane with parameters u and v.

   4. Find a parametric representation of the line that is normal at P to G_{f}.

   5. Find two linear equations in x , y , z that define the normal line as a subset of 3-dimensional space under the assumption that both of the partial derivatives of f at (a, b) are non-zero.

5. Propose a definition of the graph of a function of n variables that is consistent with the previous concepts of graph of a function when n has the special values 1 and 2.