Slice Curves & Surfaces
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) Text We begin the study of a function f of three variables x, y, and z in a rectangular domain by fixing two of the variables to produce a function of a single variable defined over an interval.
If we fix y = y_{0} and z = z_{0}, then the function f(x,y_{0},z_{0}), called the xslice function over y = y_{0}, z = z_{0}, is a function only of x defined on the interval a {leq} x {leq} b. Similarly if we fix x = x_{0} and z = z_{0}, we obtain the yslice function f(x_{0},y,z_{0}) of the one variable y, defined over the interval c {leq} y {leq} d, and fixing x = x_{0}, y = y_{0} gives the zslice function f(x_{0},y_{0},z) of the variable z defined over the interval j {leq} z {leq} k.
The collection of points (x,y_{0},z_{0},f(x,y_{0},z_{0})) is the graph of the slice function over y = y_{0}, z = z_{0}, called the slice curve for y = y_{0}, z = z_{0}.
Similarly we define the slice curve for x = x_{0}, z = z_{0} as a function of y, and the slice curve for x = x_{0}, y = y_{0} as a function of z.
By fixing one of the three variables, for example z = z_{0}, we obtain a function of the remaining two variables x and y. The graph of this function is called the z_{0}slice surface.
Similarly, we obtain the x_{0} and y_{0} slice surfaces through the point (x_{0},y_{0},z_{0}).
Demos
Slice Curves
 
In this demo, we show the slice curves of the function f(x,y,z) = x^{2}  y^{3} + z^{5}. The xslice curve is the curve obtained by fixing y = y_{0} and z = z_{0}, and it corresponds to the collection of points (x,y_{0},z_{0},f(x,y_{0},z) in 4space. The graph of this slice depends only on one parameter, x, so it is a curve. In the XSlice Curve window, we map this curve from 4space into the collection of points (x,f(x,y_{0},z_{0}) in the plane y = z = 0.
Similarly, we obtain the yslice curve by fixing x = x_{0} and z = z_{0}, and then projecting down into the plane x = z = 0; and we obtain the zslice curve by fixing x = x_{0} and y = y_{0}, projecting the curve into the plane x = y =0.

Exercises Type in the function f(x,y,z) = 2xyz/(x^{2} + y^{2} + z^{2}) What happens to slice curves that pass through the origin? You may want to change the resolution of the graph by increasing the number of x, y, and z steps.
