Changes

Other geometric equations from two dimensions will find familiar analogues in three dimensions.  For example, recall the formula for a circle centered at the origin in the plane, <math>x^2 + y^2 = r^2 \ </math>, where <math>r \ </math> is the radius of the circle.  It should come as no shock to learn the formula for a sphere in three dimensions is <math>x^2 + y^2 + z^2 = r^2 \ </math>.  Other familiar geometric curves from the plane find analogues in three dimensions as well:  just as the shape described by <math>y=x^2 \ </math> in the plane is called a '''parabola''', so the surface described by <math>z=x^2 + y^2 \ </math> is called a '''parabaloid'''.  At this time, it is not necessary to discuss all of the various surfaces which appear often - the basic forms are below, and the exercises here and in future lectures will be sufficient to introduce the student to them.

Other geometric equations from two dimensions will find familiar analogues in three dimensions.  For example, recall the formula for a circle centered at the origin in the plane, <math>x^2 + y^2 = r^2 \ </math>, where <math>r \ </math> is the radius of the circle.  It should come as no shock to learn the formula for a sphere in three dimensions is <math>x^2 + y^2 + z^2 = r^2 \ </math>.  Other familiar geometric curves from the plane find analogues in three dimensions as well:  just as the shape described by <math>y=x^2 \ </math> in the plane is called a '''parabola''', so the surface described by <math>z=x^2 + y^2 \ </math> is called a '''parabaloid'''.  At this time, it is not necessary to discuss all of the various surfaces which appear often - the basic forms are below, and the exercises here and in future lectures will be sufficient to introduce the student to them.

[[Image:Quadrics.jpg|left|900px|thumb|Some common surfaces found in three dimensional space.  In the equations given, multiplying any coordinate by a constant yields a similar surface.]]

[[Image:Quadrics.jpg|center|900px|thumb|Some common surfaces found in three dimensional space.  In the equations given, multiplying any coordinate by a constant yields a similar surface.]]

=Scalar Functions=

=Scalar Functions=