Download PDF by Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith: A Course in Higher Mathematics Volume II: Advanced Calculus

By Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith

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The loading from any element df with abscissa f is /(£) d$, and its moment about N is ( !

We differentiate both sides of (78), and obtain, since G is no longer constant: MEg£L d x + ^ y ° ) dy + - M ^ - d C = 0. e. e. when (dtp(x, y, G)jdC) dC — 0. e. a curve of the family and not the envelope; so t h a t t o obtain the envelope we must put dyj(xyyfC) dC __ ft ~""U- This equation also defines C as a function of (x, y). e. the equation of the envelope of family (IS) can be obtained by eliminating G from the two equations: V(x,y,C) = 0; » < y ° > =0. (82) As we move along the envelope, we touch different curves of family (78), each curve being defined by its value of constant C; this makes it clear why the equation of the envelope was sought in the form (78), with 0, however, taken as variable.

As we have said, a first order differential equation y' = f(v, y) (42) consists of a relationship between the coordinates (x, y) of a point and the slope y' of the tangent at this point. We assume t h a t f{x, y) is a single-valued, continuous function of (x, y). , y) is de­ «4 fined. On indicating the direction % of this tangent by an arrow passing through the corresponding point, we ffl Mo arrive at a tangent field in the plane, every tangent being associated with some point of the plane. The integral p*3 *o X curves of equation (42) are the curves 5 0 the tangents of which are the tan­ gents of the field and they may be designated the integral curves of the FIG.

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A Course in Higher Mathematics Volume II: Advanced Calculus by Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith

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