微积分教学资料chapter10.ppt

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1、Chapter 10 Parametric Equations andPolar Coordinates 10.3 Polar coordinates10.3 Polar coordinates0Polar axisP(r,)rThe point o is called the poleThe point P is represented by theordered pair(r,)and r,are call-ed polar coordinates of Pis positive if measure in the cou-nterclockwise direction from the

2、po-lar axis and negative in the clockwi-se direction(r,)(-r,)The points(r,)and(-r,)lie on the same line through o and at the same distance|r|from oExample Plot the points whose polar coordinates are given524(a)(1,)(b)(2,3)(c)(2,)(d)(3,)433Solution 5(1,)4(2,3)2(2,)34(3,)3553(1,)can be written as(1,2)

3、or(1,)or(1,)4444 5(1,)45(1,2)4 3(1,)4(1,)4The connection between polar and Cartesian coordinatesxrcosyr sinyrP(x,y)=p(r,)222yrxyarctanxExample Identify the curve by finding a Cartesian equationfor the curve(1)2(2)r=3sinrSolution2222222222(1)2the Cartesian equation is4(2)=3the Cartesian equation is3x

4、ysoxyyxysoxyxyyExample Find a polar equation for the curve representedby the given cartesian equation 222(1)(2)4xyxyx Solution22(1)()the polar equation is (2)4the polar equation is4rcosr sinsorcsccotrrcossorcos D1212oxyxy 2422 ryx1122 ryx.21,40:rD222:ayxD xyaDarD020:xyxD2:22 xyD2ocos2cos22222rrrxyx.cos20,22:rD