Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.1 Functions of Two Variables 2.1.1 Introduction 2.1.2 Linear Functions 2.1.3 Domain, Range & Function Graphs 2.1.4 Slice Curves 2.1.6 Continuity 2.4 Integration Search

Domain, Range & Function Graphs (Page: 1 | 2 )

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DOMAIN, RANGE & FUNCTION GRAPHS IN POLAR COORDINATES

The domain of a function of two variables in polar coordinates is a subset of the polar coordinate plane { (r,θ) | r ∈ R \mbox{and} 0 {leq} θ < 2π.

The range of a real-valued function f is the collection of all real numbers f(p) where p is in the domain of f.

The graph of a function of two variables is the collection of points (x,y,f(x,y)) in 3-space where (x,y) is in the domain of f. When we write the domain in polar coordinates, the graph is said to be in cylindrical coordinates.

Demos

 Domain, Range & Function Graphs This demonstration graphs a function f[r,θ] over a disc domain. By default, the domain is set to 0 ≤ r ≤ 1, -π ≤ θ ≤ π and the function is set to f[r,θ] = r2cos(2θ). In the window labeled Domain and Range, you can choose both the center and radius of a red disc domain in the xy-plane using the white and red hotspots respectively. The magenta line segment along the z-axis in this window shows the range of f[r.θ] over the red disc domain.

Exercises

• What is the range of the function f(x,y) = -x4 + 2x2 - y2 over a unit disc domain centered at the origin (i.e. the set of all points (x,y) such that 0 ≤ r ≤ 1)?
• Describe the graph of f(x,y) = 2xy/(x2 + y2) for all values of x, y other than (0,0) where the function is not defined.