By Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith
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Translated from the fourth German variation via F. Steinhardt, with an increased Bibliography.
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Extra resources for A Course in Higher Mathematics Volume II: Advanced Calculus
Isogonal trajectories. ,y,O)=0 (87) at a given angle. If the given angle is a right angle, the trajectory is called the orthogonal trajectory. We show t h a t finding an isogonal trajectory leads to integrating a first order differential equation. On eliminating C from the equations: we obtain the differential equation of the given family (87) as in : # ( * , V, 2/0 = 0. (88) We start by finding the orthogonal trajectory. e. the slopes of the tangents to the trajectory are the reciprocals, with reversed sign, of the slopes of the tangents to the given family.
7) On assigning definite values to constants Cv C2, . . , Cn, we obtain particular solutions of the equation. We obtain n equations by differentiating equation (6) or (7) (n — 1) times with respect t o x then substituting x = x0 and initial con ditions (5). I t is assumed t h a t these equations are soluble with respect to Cv G2, .. ,Cn for any given initial conditions (x0, y0, y'0, . . ,2/onl)We thus obtain the solution satisfying conditions (5). If the righthand side of equation (2) is a many-valued function, there will be several solutions of equation (7) corresponding to initial conditions (5).
We assume that the fluid flow takes place in a plane, so that a vector v, the velocity of motion, is defined at every point (x, y) of the plane. If the velocity vector depends only on the position of the point in the plane, and not on time, the motion is described as steady or established. We shall confine ourselves to this type of motion. 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.
A Course in Higher Mathematics Volume II: Advanced Calculus by Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith