By William Frederick Durand

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**Read Online or Download Aerodynamic Theory: A General Review of Progress Under a Grant of the Guggenheim Fund for the Promotion of Aeronautics PDF**

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**Additional info for Aerodynamic Theory: A General Review of Progress Under a Grant of the Guggenheim Fund for the Promotion of Aeronautics**

**Sample text**

Thus in the case of a term or terms in z, such as that above, involving both x and y, the integration of either partial derivative relative to the variable appearing in the denominator of its symbolic form, will give the complete expression for z, insofar as such terms are concerned. Suppose, however, that we have z = x2 + ys Then taking partial derivatives we have: oz BX=2x -az- -3 - y2 oy Here it is obvious that by integrating ozfo x we can only get the term x 2 while by integrating ozfoy we get only y 3 • In such case, therefore, we cannot get the complete function from either partial derivative above, but must have both and use both.

Derivatives of Hyperbolic Functions. If we express the values of sinh x, cosh x, tanh x, etc. inh x) dx - d (cosh x) dx d (tanh x)__ dx = sinh x = sech2 x 14. Illustrations of Complex Functions. 2) will be of interest, both in themselves and as showing the manner of dealing with different types of functions of this character. Later reference will be made to several of these functional forms. Hp . + Hp • Hence ·x -x-=-s+-,--y""""2 -y and x2+ yB iy X x2 + 112 iy rz- rz X cos f) r _ x2 + 11s or if we put x2 = () +y 2 = r2 .

Evidently in such case, f (0) = f (- 0). Likewise we have sin nO=- sin(- nO). SECTION 2 19 Hence in the formula for a B coefficient, the elements in the integration will form pairs equal in value and opposite in sign and thus cancelling out in the summation. Hence each B coefficient will vanish. Again since cos nO= cos(- nO), the elements in the integration for an A coefficient will form pairs equal in value and of the same sign and thus combining in the summation. The result for the entire integration from - :n: to + n will be, therefore, twice that from 0 to n.