Engineering Calculus Notes 17

Engineering Calculus Notes 17 - r,θ of P to its...

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1.1. LOCATING POINTS IN SPACE 5 x -axis y -axis z -axis P ( x,y,z ) z x y Figure 1.4: Pictures of Space In what follows, we will denote the distance between P and Q by dist( P,Q ) . Polar and Cylindrical Coordinates Rectangular coordinates are the most familiar system for locating points, but in problems involving rotations, it is sometimes convenient to use a system based on the direction and distance of a point from the origin. For points in the plane, this leads to polar coordinates . Given a point P in the plane, we can locate it relative to the origin O as follows: think of the line through P and O as a copy of the real line, obtained by rotating the x -axis θ radians counterclockwise; then P corresponds to the real number r on . The relation of the polar coordinates (
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Unformatted text preview: r,θ ) of P to its rectangular coordinates ( x,y ) is illustrated in Figure 1.5 , from which we see that x = r cos θ y = r sin θ. (1.3) The derivation of Equation ( 1.3 ) from Figure 1.5 requires a pinch of salt: we have drawn θ as an acute angle and x , y , and r as positive. In fact, when y is negative, our triangle has a clockwise angle, which can be interpreted as negative θ . However, as long as r is positive , relation ( 1.3 ) amounts to Euler’s de±nition of the trigonometric functions ( Calculus Deconstructed , p. 86). To interpret Figure 1.5 when r is negative , we move | r | units in the opposite direction along ℓ . Notice that a reversal in the...
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