multivariable_10_Polar_Coordinates,_Parametric_Equations

multivariable_10_Polar_Coordinates,_Parametric_Equations -...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 10 Polar Coordinates, Parametric Equations aC d iae Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian ) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinates a point in the plane is identified by a pair of numbers ( r, ). The number measures the angle between the positive x-axis and a ray that goes through the point, as shown in figure 10.1; the number r measures the distance from the origin to the point. Figure 10.1 shows the point with rectangular coordinates (1 , 3) and polar coordinates (2 , / 3), 2 units from the origin and / 3 radians from the positive x-axis. 3 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2 , / 3) . . . . . . . Figure 10.1 Polar coordinates of the point (1 , 3). 217 218 Chapter 10 Polar Coordinates, Parametric Equations Just as we describe curves in the plane using equations involving x and y , so can we describe curves using equations involving r and . Most common are equations of the form r = f ( ). EXAMPLE 10.1 Graph the curve given by r = 2. All points with r = 2 are at distance 2 from the origin, so r = 2 describes the circle of radius 2 with center at the origin. EXAMPLE 10.2 Graph the curve given by r = 1 + cos . We first consider y = 1 + cos x , as in figure 10.2. As goes through the values in [0 , 2 ], the value of r tracks the value of y , forming the cardioid shape of figure 10.2. For example, when = / 2, r = 1 + cos( / 2) = 1, so we graph the point at distance 1 from the origin along the positive y-axis, which is at an angle of / 2 from the positive x-axis. When = 7 / 4, r = 1 + cos(7 / 4) = 1 + 2 / 2 1 . 71, and the corresponding point appears in the fourth quadrant. This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated. 1 2 / 2 3 / 2 2 ................................... .. .. .. . . . . . . . . .. .. ... ............................................................... . . .. .. . . .. . .. . . . .. .. .. .... ............................... . . . . . . . . . . . ....
View Full Document

This note was uploaded on 12/01/2011 for the course MATH 305 taught by Professor Guichard during the Fall '11 term at Whitman.

Page1 / 16

multivariable_10_Polar_Coordinates,_Parametric_Equations -...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online