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## Calculus

by: Otilia Murray I

14

0

5

# Calculus MAT 021B

Otilia Murray I
UCD
GPA 3.88

Qinglan Xia

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COURSE
PROF.
Qinglan Xia
TYPE
Class Notes
PAGES
5
WORDS
KARMA
25 ?

## Popular in Mathematics (M)

This 5 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 021B at University of California - Davis taught by Qinglan Xia in Fall. Since its upload, it has received 14 views. For similar materials see /class/187344/mat-021b-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
11 5 Polar coordinates Given a point P in the plane how to represent it mathematically o Cartesian coordinates O U r 7 axis U y 7 axis 0 Polar coordinate O U polar axis 0 A O 4 u the right Mr 0 Usually the polar 39 De nition of Polar Coordinates Oliglll mic For any point P in the plane let 0 7 be the directed distance from O to P o 5 be the directed angle from the polar axis to the line OP 0 H gt p r o S is positive when measured counterclockwise mm m o S is negative when measured clockwise n 1035 To mm P0lar coordiunlcs rm the plane We slan wiih an he angle 5 15 usually measured 1 m lg caucd the p01 and an initial my en the point P is represented by the ordered pair 7 H and 79 are called polar coordinates of P Note that if7 0 then 0 5 always represent the pole O for any 9 Negative r There are occasi s when we wish to allow 7 to be negative That is why we use directed distance in de ning P7 S Als it is convenient to agree that 77 represents the same point as 7 S 7r That is 77 S and 7 5 lie on the same line 0 at the same distance H from 0 but on opposite sides of FIGURE 1037 Polar coordinates can have negative Tvalues Example Plot each of the following polarform poinm 447973979panacea E 4 F 4 g a 4 A 0 Note that T 6 T 6 27f7 6 7 27f 77 6 7r all represenm the same point Or in general for any integer n T 6 27m and 77 6 2n 17r all represent the same point P T 6 Therefore polar representation is not uni ue Example Find all the polar coordinates of the point P2 7r Relation between polar and Cartesian Coordinates When we use both polar and cartesian coordinates in a plane we place any 7 o the pole 0 be the same point as the origin 0 m mm o the polar axis be the positive xiaxis Then any point P in the plane can be represented by r 6 in polar form or x y in cartesian form FIGURE 1041 Thcusunl way In relate polar and Cartesian coordjnzlcs 0 To change from polar r 6 to Cartesian x we use x r cos 6 yrsin6 Example Convert the point 72 from polar to Cartesian coordinates 0 To change from Cartesian x y to polar r 6 we use arctan if x y 0 6 7 if x 0 y gt 0 g if x 0y lt 0 r jm y2 lN Note the sign of r is chosen to ensure that the point P lie in the appropriate quadrant 0 sign is if P is in quadrant I or IV 0 sign is ifP is in quadrant II or III Example Convert the point lag from Cartesian to polar coordinates Example What curve is represented by or3 or 3 09 Example Graph the sets of points Whose polar coordinates satisfy the following conditions olSrSZand gg6g7r o 2 rlt3and6 o r S 0 0 g o g g 0 g 317 no restriction on 7 Example Find a polar equation for the circle 952 y 22 4 Example Find an equivalent Cartesian equation for the following polar equations 0 7 cos 0 5 o 7 2 6 sin 0 O 4 r 3 cos 0 2 Sin 0 o r2lsin0 o r cos 26

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