Section 12.5Functions 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)

222),,( zyxzyxfT

1222 zyx

• Suppose w = f(x,y,z)• Now the inputs are ordered triples of numbers• We can’t graph because we don’t have a 4th 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?

222

1000),,(

zyxzyxfB

• 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?

)sin(),,(

),,(

),,( 22

zyxzyxg

xzzyxh

zxzyxf