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Download A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.) PDF

By V. I. Smirnov and A. J. Lohwater (Auth.)

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Extra resources for A Course of Higher Mathematics. Volume II

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Noting that continuous loading is absent here and that Mi = O, we have from equations (28) and (29): E0 — P0 = —P (reaction force); M0 = IP{ (reac­ tion couple). 2. We find the curve of bending of a girder, rest­ MA ing on two supports A and B (Fig. 21) and subjected to a head of water whose level is opposite the upper support (dam). The forces acting on the girder here amount to (1) the continuously distributed head of water, and (2) the reactions at the supports. &B Let b be the width of the girder and Q be the weight F I G .

We use the same method of differentiation for (72) as for the Clairaut equation. We write y' = p, so t h a t the equation becomes y = xi(p) - p]-^- + rite)X + a n ( i obtain its general solution in the form: x = Vi{p)C + V2{p).

E. t h a t the projections of vector v(x, y) on the coordinate axes are the partial derivatives du(xf y)/dx and du(xiy)jdy of some function u(x,y). The curves of the family u (x, y) = C (90) are described in this case as equipotential lines. The lines, the tangents to which have, at every point, the same direction as the vector Y(X, y)> are called stream lines and give the trajectories of the mov­ ing particles. We show that the stream lines form the orthogonal trajectories of the family of equipotential lines.

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