tailieunhanh - Giáo trình toán kỹ thuật 3

Các mô hình và các giả thiết toán học trong hầu hết các trường hợp đều tìm cách tiếp cận một cách gần đúng nhất với bản chất vật lý của hiện tượg và đơn giản hoá các vấn đề kỹ thuật trong các điều kiện nhất định. | For cylindrical coordinates. -2- 1 5 _ 1 7 7 p dp V dp p2 dip7 dz2 1 3 Ể d2f 1 32 d2f p dp dp2 p- 3ự 2 dz2 1-84 1-85 except on the z-axis For spherical coordinates r2 sin 0 30 1-86 2 df . 37 . cot idf 1 37 4 I-TT 1 SY rdr dr2 r2 dd r2 do2 r2 sin- e 3ự 2 1-87 except again on the z-axis. We have already seen in Sec. LIO that the Laplacian of a vector ỈÌ in Cartesian coordinates is itself a vector whose components arc the Laplacians of Bx ổz. Then 1-88 is an identity in Cartesian coordinates. With other coordinates V R ist by definition the vector whose components are those of F 7 fì - r X F X fi and not the sum of the l-aplacians of Bx B2 Be V2B F P B - FX FXB 1-89 19 CHƯƠNG 2. HÀM BIÉN PHỨC THE DERIVATIVE IN THE COMPLEX PLANE THE CAUCHY-RIEMANN EQUATIONS We have already introduced the complex variable z i iy where z and y are variable. We new define another complex variable w U-H v so that for each value of 2 there corresponds a value of w f z . From all of the possible complex functions that we might invent we will focus on those functions where for each z there is one and only one value of w. These functions are single-valued. They differ from functions such Figure The complex function w 2 . as the square root logarithm and inverse sine and cosine where there are multiple answers for each z. These multivalued functions do arise in various problems. However they are beyond the scope of this book and we shall always assume that we are dealing with single-valued functions. A popular method for representing a complex function involves drawing some closed domain in the z-plane and then showing the corresponding domain in the w-plane. This procedure is called mapping and the 2-plane illustrates the domain of the function while the w-plane illustrates its image or range. Figure shows the 2-plane arid w-plane for w 22 a pie-shaped wedge in the Z plane maps into a semicircle on the w-plane. 20 Although the requirement that a complex function be single-valued is