Calculus

Download e-book for kindle: All You Wanted to Know About Mathematics but Were Afraid to by Louis Lyons

By Louis Lyons

ISBN-10: 0521436001

ISBN-13: 9780521436007

Physics and engineering scholars want a transparent realizing of arithmetic for you to remedy an enormous array of difficulties posed to them in coursework. regrettably, in lots of textbooks, mathematical proofs and methods imprecise a primary knowing of the actual ideas. In a transparent and didactic demeanour, this booklet explains to the coed why specific complex mathematical thoughts are invaluable for fixing sure difficulties. the purpose is to exhibit a deeper appreciation of mathematical tools which are acceptable to physics and engineering via a dialogue of a variety of genuine actual difficulties. the themes coated contain simultaneous equations, third-dimensional geometry and vectors, complicated numbers, differential equations, partial derivatives, Taylor sequence, and Lagrange multipliers.

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Additional resources for All You Wanted to Know About Mathematics but Were Afraid to Ask (Mathematics for Science Students, Volume 1)

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On the other hand, minimization of the surface energy alone would lead to a spherical drop, which minimizes the surface area and, therefore, the total surface energy for a drop of a given volume. The actual shape of the drop is somewhere between these two extremes depending upon the physical parameters g, ρ, σ , and α. Because we seek to minimize the definite integral E[u(r)] involving an unknown function u(r), this is a problem in calculus of variations. Again, observe that the variational form of the problem arises naturally from a consideration of the physical problem from first principles.

Integrating again, we have u(x) = c1 x + c2 , which as expected is a straight line. Applying the boundary conditions u(x0 ) = u0 and u(x1 ) = u1 leads to the solution u(x) = u1 − u0 (x − x0 ) + u0 . 20) and evaluating the definite integral. Although it may be stretching the point somewhat, we can go so far as to say that the fundamental building blocks of Euclidean geometry are based on variational principles: • • • • straight line → shortest distance between two points. circle → shortest curve enclosing a given area.

2 provides a clearer understanding of the underlying principles through use of arguments from differential calculus, the derivation given in this section using variational notation is more concise. Moreover, it highlights the parallels with differential calculus of extrema of functions as follows: • Taking the variation of the functional and setting it equal to zero, that is, δI = 0, is equivalent to differentiating I[u] with respect to and setting equal to zero. • The variation of u(x), δu, plays the same role as η(x) accounting for the variation from curve to curve.

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All You Wanted to Know About Mathematics but Were Afraid to Ask (Mathematics for Science Students, Volume 1) by Louis Lyons


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