]> 31.2 Expressing Surfaces Parametrically or in Appropriate Form for Integration

31.2 Expressing Surfaces Parametrically or in Appropriate Form for Integration

To handle qualitative descriptions you need to be able to write an equation or parametric representation of the surface in question. The important surfaces that you should be prepared to handle in this way are: planes, spherical and ellipsoidal surfaces, cylindrical surfaces and conic surfaces.

We very briefly review each of these.

A plane is described by a linear equation

$a x + b y + c z = d$

You can always solve for one variable in terms of the other two. The coefficients are then determined by the conditions given on the plane.

A spherical surface of radius $A$ centered at $( x ' , y ' , z ' )$ consists of the points of space obeying the equation

$( x − x ' ) 2 + ( y − y ' ) 2 + ( z − z ' ) 2 = A 2$

A general ellipsoid has surface characterized by

$( ( x − x ' ) A ) 2 + ( ( y − y ' ) B ) 2 + ( ( z − z ' ) C ) 2 = 1$

A cylinder with radius $A$ with axis parallel to the z-axis starting at $z = B$ and ending at $z = C$ has surface represented by the equation

$( x − x ' ) 2 + ( y − y ' ) 2 = A 2 , for B < z < C$

A cone with center line parallel to the z-axis has surface described by

$( x − x ' ) 2 + ( y − y ' ) 2 = A ( z − z ' ) 2$