Labware - MA35 Multivariable Calculus - Three Variable Calculus
 MA35 Labs 3 » Three Variable Calculus Contents3.1 Functions of Three Variables 3.1.1 Linear Functions 3.1.3 Slice Curves & Surfaces 3.1.5 Continuity 3.4 Integration Search

Slice Curves & Surfaces (Page: 1 | 2 | 3 )

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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.

The collection of points (x,y0,z0,f(x,y0,z0)) is the graph of the slice function over y = y0, z = z0, called the slice curve for y = y0, z = z0.

By fixing one of the three variables, for example z = z0, we obtain a function of the remaining two variables x and y. The graph of this function is called the z0-slice surface.

Demos

 Slice Curves In this demo, we show the slice curves of the function f(x,y,z) = x2 - y3 + z5. The x-slice curve is the curve obtained by fixing y = y0 and z = z0, and it corresponds to the collection of points (x,y0,z0,f(x,y0,z) in 4-space. The graph of this slice depends only on one parameter, x, so it is a curve. In the X-Slice Curve window, we map this curve from 4-space into the collection of points (x,f(x,y0,z0) in the plane y = z = 0. Similarly, we obtain the y-slice curve by fixing x = x0 and z = z0, and then projecting down into the plane x = z = 0; and we obtain the z-slice curve by fixing x = x0 and y = y0, projecting the curve into the plane x = y =0.

Exercises

• Type in the function f(x,y,z) = 2xyz/(x2 + y2 + z2) 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.