By Edwin Bidwell Wilson
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Additional resources for Advanced calculus; a text upon select parts of differential calculus, differential equations, integral calculus, theory of functions; with numerous exercises
Note that r ≥ 0, 0 ≤ θ < 2π, ρ ≥ 0 and 0 ≤ φ ≤ π. Also, θ is undefined when ( x, y) = (0, 0), and φ is undefined when ( x, y, z) = (0, 0, 0). 13 This “standard” definition of spherical coordinates used by mathematicians results in a left-handed system. For this reason, physicists usually switch the definitions of θ and φ to make (ρ , θ , φ) a right-handed system. 48 CHAPTER 1. 31. Convert the point (−2, −2, 1) from Cartesian coordinates to (a) cylindrical and (b) spherical coordinates. 23 9 = 3, φ = cos−1 1 3 −2 −2 = tan−1 (1) = 5π 4 , since y = −2 < 0.
3 Spherical coordinates Both θ and φ are measured in radians. Note that r ≥ 0, 0 ≤ θ < 2π, ρ ≥ 0 and 0 ≤ φ ≤ π. Also, θ is undefined when ( x, y) = (0, 0), and φ is undefined when ( x, y, z) = (0, 0, 0). 13 This “standard” definition of spherical coordinates used by mathematicians results in a left-handed system. For this reason, physicists usually switch the definitions of θ and φ to make (ρ , θ , φ) a right-handed system. 48 CHAPTER 1. 31. Convert the point (−2, −2, 1) from Cartesian coordinates to (a) cylindrical and (b) spherical coordinates.
12 the height h of the parallelepiped is u cos θ , and not − u cos θ , because the vector u is on the same side of the base parallelogram’s plane as the vector v × w (so that cos θ > 0). Since the volume is the same no matter which base and height we use, then repeating the same steps using the base determined by u and v (since w is on the same side of that base’s plane as u × v), the volume is w · (u × v). 12) will give you the negative of the volume of the parallelepiped. 15. If vectors u, v, w in R3 represent any three adjacent sides of a parallelepiped, then the volume of the parallelepiped is | u · (v × w) |.