Download e-book for iPad: Mathematical handbook of formulas and tables by Seymour Lipschutz, Murray Spiegel, John Liu

By Seymour Lipschutz, Murray Spiegel, John Liu

ISBN-10: 0071548564

ISBN-13: 9780071548564

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Extra resources for Mathematical handbook of formulas and tables

Example text

If b = a/4, the curve is that of Fig. 9-3. Fig. 20. Parametric equations: { x = aφ − b sin φ y = a − b cos φ This is the curve described by a point P at distance b from the center of a circle of radius a as the circle rolls on the x axis. If b < a, the curve is as shown in Fig. 9-10 and is called a curtate cycloid. If b > a, the curve is as shown in Fig. 9-11 and is called a prolate cycloid. If b = a, the curve is the cycloid of Fig. 9-2. Fig. 9-10 Fig. 21. Parametric equations: ⎨ ⎩y = a sin φ This is the curve described by endpoint P of a taut string PQ of length a as the other end Q is moved along the x axis.

Fig. 8. 9. 10. 11. Arc length of entire curve = 6a This is a curve described by a point P on a circle of radius a/4 as it rolls on the inside of a circle of radius a. 28 Fig. 12. 13. 14. Arc length of curve = 16a This is the curve described by a point P of a circle of radius a as it rolls on the outside of a fixed circle of radius a. 32). Fig. 15. Equation: y = a x /a − x /a x (e + e ) = a cos h 2 a This is the curve in which a heavy uniform chain would hang if suspended vertically from fixed points A and B.

Fig. 16. Equation: r = a cos 3u The equation r = a sin 3u is a similar curve obtained by rotating the curve of Fig. 9-6 counterclockwise through 30° or p/6 radians. In general, r = a cos nu or r = a sin nu has n leaves if n is odd. Fig. 17. Equation: r = a cos 2u The equation r = a sin 2u is a similar curve obtained by rotating the curve of Fig. 9-7 counterclockwise through 45° or p/4 radians. In general, r = a cos nu or r = a sin nu has 2n leaves if n is even. Fig. 18. Parametric equations: ⎧ ⎛ a + b⎞ ⎪⎪x = (a + b) cosθ − b cos ⎜⎝ b ⎟⎠ θ ⎨ ⎪y = (a + b)sin θ − b sin ⎛⎜ a + b⎞⎟ θ ⎝ b ⎠ ⎪⎩ This is the curve described by a point P on a circle of radius b as it rolls on the outside of a circle of radius a.

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Mathematical handbook of formulas and tables by Seymour Lipschutz, Murray Spiegel, John Liu


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