Changes

Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.   

Any line in the xy plane can be described as <math>y=ax+b \ </math>, where <math>a \ </math> is the "slope" and <math>b \ </math> is the y-intercept.  When we raise the number of dimensions, we see how similar the equation of a plane is in three-space: <math>z=ax+by+c \ </math>, where <math>a \ </math> is the slope along the x-axis, etc.   

Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can descrive its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.

Describing a line in three dimensions is trickier, and requires what is called a '''parametric equation'''.  This method of describing lines relies on some variable which isn't a coordinate, like time, to describe a point as it travels through space.  For example, if a point is at some location <math>(x_0, y_0, z_0) \ </math> at time <math>t=0 \ </math>, and then over 1 second travels <math>a \ </math> feet in the x direction, <math>b \ </math> feet in the y direction, and <math>c \ </math> feet in the z direction, we can describe its location at any time <math>t \ </math> as <math>(x_0+at, y_0+bt, z_0+ct) \ </math>.  As it travels, this point traces out a line.

 

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 exercises will be sufficient to introduce the student to them.

=Scalar Functions=

=Scalar Functions=