 By J. Demailly

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Extra info for Complex Analytic and Differential Geometry

Sample text

39). 43) where w(t) is a function of time. 33) are given by a = 2 w(t), b = -3 w(t); c = -w(t), d = 3 w(t). 36), we find that the rate of rotation is given by Jl = —w(t), and the rate of expansion is given by a — 5 w(t). 47) equal to zero. 42). 49) are, in fact, identical. 1 Material lines. A collection of point particles distributed over a line in a threedimensional flow defines a material line. Explain why, if the flow is steady, a material line that lies on a streamline at a certain time will remain on the streamline at all times.

ZJ where a, 6, and c are three constants with appropriate dimensions. In this case, the fluid moves along the x axis with velocity that depends on the y coordinate alone. 3) The solution of these equations is readily found to be X(t) = X(t = 0) + (aY2 + bY + c) t, y(t) = y(t = o), z(t) = z(t = o), (L5>4) where -X" (O), V(O), and Z(O) are the coordinates of a point particle at the initial instant, t = O. 1) may not be found by analytical methods, and the use of numerical methods will be imperative.

The qualifier explicit emphasizes that the new position of the point particle is computed in terms of the old position at a single stage by means of multiplications. 6). Accordingly, the error in the position of the point particle after it has travelled for the time interval At, will be on the order of At 2 . Based on the value of this exponent of At, we say that the explicit Euler method carries a stepwise error of second order with respect to the size of the time step. If Nsteps steps are executed from time t = O to time t = t/^ na /, the stepwise error will accumulate to an amount that is comparable to the product Nsteps x At 2 .