Section 17.2

Parametric Surfaces

Describing complex shapes by warping a 2D sheet.

Introduction

Just as a space curve is described by a function of one parameter , a surface requires two parameters, usually called and .

As varies over a region D in the parameter plane, the tip of the vector sweeps out a surface S in space.

Holding one parameter constant gives a "Grid Curve". The derivatives are tangent vectors to these curves:

The cross product gives a vector normal to the surface:

Identify the surface given by .

1. Inspect Components:

, , .

2. Eliminate Parameter u:

3. Interpret:

This is a cylinder of radius 2 centered on the y-axis.

Find a normal vector to the paraboloid at (1,2,5).

1. Partials:

2. Evaluate at Point:

Given (1,2,5), we have .

3. Cross Product:

For a sphere of radius a, . Find .

1. Known Result:

The cross product of spherical basis vectors is classic.

2. The Magnitude:

Concept:

This is the "Jacobian" term for surface area integrals on a sphere!