Integration with variables Notes_Part_12

# Integration with variables Notes_Part_12 - Center.1...

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Center: . 1 Radius: . 5 Center: . 2 + i Radius: 1 Center: 1 + i Radius: 1 Center: 2 + 3 i Radius: 3 2 4 . 2426 Center: . 2 + i Radius: 1 . 2806 Center: . 1 + . 3 i Radius: . 9487 Center: . 1 + . 1 i Radius: 1 . 1045 Center: . 2 + . 1 i Radius: 1 . 2042 Figure 7.21. Airfoils Obtained from Circles via the Joukowski Map. rest of the ζ plane, as do the images of the (nonzero) points inside the unit circle. Indeed, if we solve (7.67) for z = ζ ± r ζ 2 1 , (7 . 68) we see that every ζ except ± 1 comes from two diFerent points z ; for ζ not on the critical line segment [ 1 , 1], one point lies inside and and one lies outside the unit circle, whereas if 1 < ζ < 1, the points lie on the unit circle and on a common vertical line. Therefore, (7.67) de±nes a one-to-one conformal map from the exterior of the unit circle b | z | > 1 B onto the exterior of the unit line segment C \ [ 1 , 1]. Under the Joukowski map, the concentric circles | z | = r n = 1 are mapped to ellipses with foci at ± 1 in the ζ plane; see ²igure 7.20. The eFect on circles not centered at the origin is quite interesting. The image curves take on a wide variety of shapes; several examples are plotted in ²igure 7.21. If the circle passes through the singular point z = 1, then its image is no longer smooth, but has a cusp at ζ = 1; this happens in the last 6 of the ±gures. Some of the image curves have the shape of the cross-section through an airplane wing or airfoil . Later, we will see how to construct the physical ³uid ³ow around such an airfoil, a result that was a critical step in early aircraft design. Composition and the Riemann Mapping Theorem

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Integration with variables Notes_Part_12 - Center.1...

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