By Snieder R.
Unlike conventional textbooks for college students and pros within the actual sciences, this publication offers its fabric within the type of difficulties. the second one variation comprises new chapters on dimensional research, variational calculus, and the asymptotic review of integrals. The ebook can be utilized through undergraduates and lower-level graduate scholars. it could function a stand-alone textual content, or as a resource of difficulties and examples to enrich different textbooks. First version Hb (2001): 0-521-78241-4 First variation Pb (2001): 0-521-78751-3
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Additional info for A Guided Tour of Mathematical Methods: For the Physical Sciences, Edition: 2ed.
Fig. 3 The motion of a bouncing ball that loses energy with every bounce. To visualize the motion of the ball better, the ball is a given a constant horizontal velocity that is conserved during the bouncing. 3 Bouncing ball 25 Problem a A ball moves upward from the level z = 0 with velocity v and is subject to a constant gravitational acceleration g. Determine the height the ball reaches and the time it takes for the ball to return to its starting point. At this point we have determined the relevant properties for a single bounce.
3 that is deﬁned by inﬁnitesimal increments dr , dθ, and dϕ. Give an alternative derivation of this expression for d V that is based on geometric arguments only. 42 Spherical and cylindrical coordinates z-axis r sinq d j rdq dq dj dr r y-axis x-axis Fig. 3 Deﬁnition of the geometric variables for an inﬁnitesimal volume element dV . In some applications one wants to integrate over the surface of a sphere rather than over a volume. For example, if one wants to compute the cooling of the Earth, one needs to integrate the heat ﬂow over the Earth’s surface.
Make suitable substitutions in this result to show that the Taylor expansion around an arbitrary point (x, y) is given by ∂f ∂f (x, y) h x + (x, y) h y ∂x ∂y 1 ∂2 f ∂2 f 2 (x, y) h x h y + (x, y) h + x 2 ∂x2 ∂ x∂ y 1 ∂2 f + (x, y) h 2y + · · · . 10). This series hides an intriguing result. 10) suggest that a function f (x) is speciﬁed for all values of its argument x when all the derivatives are known at a single point x = 0. This means that the global behavior of a function is completely contained in the properties of the function at a single point.