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# 9 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

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Contents CHAPTER 9 9.1 9.2 9.3 9.4 CHAPTER 10 10.1 10.2 10.3 10.4 10.5 CHAPTER 11 11.1 11.2 11.3 11.4 11.5 CHAPTER 12 12.1 12.2 12.3 12.4 CHAPTER 13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Polar Coordinates and Complex Numbers Polar Coordinates 348 Polar Equations and Graphs 351 Slope, Length, and Area for Polar Curves 356 Complex Numbers 360 Infinite Series The Geometric Series Convergence Tests: Positive Series Convergence Tests: All Series The Taylor Series for ex, sin x, and cos x Power Series Vectors and Matrices Vectors and Dot Products Planes and Projections Cross Products and Determinants Matrices and Linear Equations Linear Algebra in Three Dimensions Motion along a Curve The Position Vector 446 Plane Motion: Projectiles and Cycloids 453 Tangent Vector and Normal Vector 459 Polar Coordinates and Planetary Motion 464 Partial Derivatives Surfaces and Level Curves 472 Partial Derivatives 475 Tangent Planes and Linear Approximations 480 Directional Derivatives and Gradients 490 The Chain Rule 497 Maxima, Minima, and Saddle Points 504 Constraints and Lagrange Multipliers 514

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C H A P T E R 9 Polar Coordinates and Complex Numbers Up to now, points have been located by their x and y coordinates. But if you were a flight controller, and a plane appeared on the screen, you would not give its position that way. Instead of x and y, you would read off the direction of the plane and its distance. The direction is given by an angle 6. The distance is given by a positive number r. Those are the polar coordinates of the point, where x and y are the rectangular coordinates. The angle 6 is measured from the horizontal. Suppose the distance is 2 and the direction is 30" or 4 6 (degrees preferred by flight controllers, radians by mathemati- cians). A pilot looking along the x axis would give the plane's direction as "11 o'clock." This totally destroys our system of units, by measuring direction in hours. But the angle and the distance locate the plane. How far to a landing strip at r = 1 and 8 = - n/2? For that question polar coordi- nates are not good. They are perfect for distance from the origin (which equals r), but for most other distances I would switch to x and y. It is extremely simple to determine x and y from r and 8, and we will do it constantly. The most used formulas in this chapter come from Figure 9.1-where the right triangle has angle 6 and hypotenuse r. The sides of that triangle are x and y: x = r cos 8 and y = r sin 8. (1) The point at r = 2, 8 = 4 6 has x = 2 cos(n/6) and y = 2 sin(n/6). The cosine of n/6 is J 5 / 2 and the sine is f. So x = \$ and y = 1. Polar coordinates convert easily to xy coordinates-now we go the other way. Always x2 + y2 = r2. In this example (&)2 + = (2)2. Pythagoras produces r from x and y. The direction 8 is also available, but the formula is not so beautiful: r = J w and tang= - Y and(a1most) ~ = t a n - ' Y . (2) X X Our point has y/x = I/&. One angle with this tangent is 8 = tan-' ( 1 1 8 ) = n/6.
9.1 Polar Coordinates Fig. 9.1 Polar coordinates r, 8 and rectangular coordinates x = r cos 9, y = r sin 8.

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9 - Contents CHAPTER 9 9.1 9.2 9.3 9.4 Polar Coordinates...

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