Section 12.5 Functions of Three Variables

Consider temperature at a point in space –It takes 3 coordinates to determine a point in space, (x, y, z), then T = f(x, y, z) –Thus we have a function of 3 variables Suppose –Then there is only one place where the temperature is 0 (at (0, 0, 0)) –There are an infinite number of points that have a temperature of 1°, namely all the points on the sphere –We could continue to create these spheres, each corresponding to a different temperature –Each sphere is a level surface of f(x, y, z)

Suppose w = f(x,y,z) Now the inputs are ordered triples of numbers We can’t graph because we don’t have a 4 th dimension We can graph level surfaces of f f(x,y,z) = k is an equation of 3 variables, x, y, and z, that defines a surface in 3 space on which the function takes an output value of k

For example, consider a point light source in space –It’s brightness is inversely proportional to the square of the distance from the source is given by (using 1000 as the constant) –Let’s take a look a the plot in MAPLE –Now find a level surface where f(x,y,z) = 10 What kind of shape does it have? What if the value is k instead of 10?

What is the relationship between functions of 2 variables and functions of 3 variables? –A single surface is used to represent a 2 variable function, f(x,y) –A family of surfaces is used to represent a 3 variable function, g(x,y,z) For example, what do the level surfaces of the following functions look like?