By Edwin Bidwell Wilson

It is a replica of a ebook released sooner than 1923. This publication can have occasional imperfections comparable to lacking or blurred pages, bad images, errant marks, and so forth. that have been both a part of the unique artifact, or have been brought via the scanning method. We think this paintings is culturally very important, and regardless of the imperfections, have elected to deliver it again into print as a part of our carrying on with dedication to the renovation of published works around the globe. We take pleasure in your figuring out of the imperfections within the renovation method, and desire you take pleasure in this worthwhile booklet.

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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

Example text

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) |.

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