The prototypical example of a function of several variables is the Temperature at each point in a closed room. Each point in the room can be labeled with its three coordinates in a given coordinated system with basis elements i,j,k. Here are some examples of functions of several variables defined in maple: |

> T := (x,y,z) -> x^2+y^2+3*z^2;

2 2 2 T := (x,y,z) -> x + y + 3 z

This could represent a temperature field in a room. For example the temperature at (0,1,-1) = j-k is |

> T(0,1,-1);

4

The volume V of a circular cylinder of radius r and hight h is also a function of several variables, but in this case of only 2. |

> V := (r,h) -> Pi*r^2*h;

2 V := (r,h) -> Pi r h

As in the case of a vector function we can go back from an expression to the corresponding function by "unapply"-ing. For example: |

> g := unapply(sqrt(9-x^2-y^2),x,y);

2 2 1/2 g := (x,y) -> (9 - x - y )

and we can now evaluate at different (x,y)'s with: |

> z1 := g(2,1); z2 := g(-1,Pi);

1/2 z1 := 4 2 1/2 z2 := (8 - Pi )

Functions of two variables can be visualized as surfaces in 3D. For example |

> z := g(x,y);

2 2 1/2 z := (9 - x - y )

can be seen as providing a rule for attaching a stick of length z to each point (x,y) in the circle of radius 3 centered at 0 on the xy-plane. We can look at it with: |

> plot3d(z,x=-3..3,y=-3..3,axes=framed,color=green);

## Level CurvesThe level curves of a function of two variables are the curves where the function takes a constant value. The level curves have equations: |

> z := x*exp(-x^2-y^2);

2 2 z := x exp(- x - y )

> with(plots):

> contourplot(z,x=-2..2,y=-2..2,grid=[49,49], color = z);

Link to the commands in this file

Carlos Rodriguez <carlos@math.albany.edu> Last modified: Mon Oct 7 21:07:41 EDT 1996