By J.E. Thompson

204 Pages

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12 3 4 5 + + not. Let Un = # of shaded squares ( U = uncertain) Cn = #of squares whose vertices all have X's (C = certain) An = 4 (:2 • Cn + ~2 · ~ · Un) (A = approximation) Find A1o. Find A2o. 3 j Definition of Certain Simple Integrals 35 the integral JD dx dy which is based on the subdivision by lines x = ± 1!.. , y = ± '!. or on any refinement of this subdivision n n differs from An by at most 2n- 2 Un. What accuracy (how many decimal places) does this estimate guarantee for the approximation A 1 o "' 1r?

What is a constant 0-form? 2 Integration If the force field is not constant, then finding the amount of work required for a given displacement requires a process of integration. The essential idea is that the 1-form (1) I I (@_ I I I / I --- A(x, y, z) dx + B(x, y, z) dy + C(x, y, z) dz which describes the force field gives the approximate amount of work required for small displacements. The amount of work required for a displacement which is not small can then be described as a limit of sums of values of (1).

0, 1) agrees with the orientation PoP1P2P3 where Po is the base of the thumb, P 1 the tip of the thumb, P2 the tip of the index finger, and P3 the tip of the third finger of a right hand held in the natural position so that these points are non-coplanar. expressing xyz directly in terms of rst and then forming the pullback or by first expressing dx dy dz as a 3-form in uvw and then expressing this 3-form as a 3-form in rst. 3, by writing out the computations explicitly in terms of the 18 coefficients a, b, c, a', ...