Read e-book online A Course in Higher Mathematics Volume II: Advanced Calculus PDF

By Smirnov, Vladimir Ivanovič; Sneddon, Ian Naismith

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FIG. 16 FIG. 17 along the rule from thi3 hole. One leg of a small tripod device is located at the point corresponding to R, whilst the other two legs are on the paper. On shifting the tripod along the scale at points Mlf M2 etc. in accordance with the variation of R, we do not alter the direction of the tangent a t these points; hence we obtain the required curve. We now give another method of graphical integration of equation (12), providing an approximation to the integral curve in the form of a step line.

The family of isoclines (66) is at the same time the family of the general solution of (65). We now indicate a second method of obtaining the general solution of equation (65), whereby the singular solution of the equation is 8] 29 CLAIRATJT'S EQUATION found, as well as its general solution. We use the notation y' = p, and re-write (65): y = xp + (X).

We obtain a slope which is the reciprocal, with reversed sign, of slope (91). Hence it follows that the equipotential lines and the stream lines are orthogonal to each other. 12] ISOGONAL TRAJECTORIES 39 If a family of curves is a family of equipotential lines, its orthogonal trajectories form the family of corresponding stream lines, and vice versa. In the case of a plane electrostatic field, the lines of force represent the orthogonal trajectories of the family of equipotential lines. Example.

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


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