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Mathematical Tools Appendix Mathematical Tools Contents A.1 Some Vector Calculus Results.............................................................. 389 A.1.1 Cartesian Index Notation.......................................................... 389 A.1.2 Derivatives of Fundamental Solutions of Laplace’s Equation................... 392 A.1.3 The Divergence Theorem and Some Relevant Uses.............................. 393 A.1.4 Stokes’ Theorem and Some Relevant Uses....................................... 405 A.1.5 Change of Variables Theorem..................................................... 411 A.1.6 Field Quantities and Their Rate of Change....................................... 411 A.1.7 Time Differentiation of Spatial Integrals.......................................... 412 A.2 Useful Tools from Complex Analysis...................................................... 416 A.2.1 Basic Properties.................................................................... 417 A.2.2 The Cauchy Integral and Residue Theorem...................................... 420 A.2.3 Conformal Mapping............................................................... 425 A.2.4 The Joukowski and Kármán–Trefftz Airfoils..................................... 431 A.2.5 The Schwarz–Christoffel Transformation......................................... 436 A.3 Mathematical Results for the Infinitely-Thin Plate......................................... 443 A.3.1 Notes on an Important Factor...................................................... 443 A.3.2 Properties of Chebyshev Polynomials............................................. 444 A.3.3 Contour Integrals of Interest....................................................... 452 A.1 Some Vector Calculus Results This section provides a number of useful identities from vector calculus. A.1.1 Cartesian Index Notation Before we go any further, let’s review the rules of index notation, a useful tool in proving identities. The convention we use here is a stripped-down version of the actual convention, which, for example, distinguishes covariant and contravariant © Springer Nature Switzerland AG 2019 389 J. D. Eldredge, Mathematical Modeling of Unsteady Inviscid Flows, Interdisciplinary Applied Mathematics 50, https://doi.org/10.1007/978-3-030-18319-6 390 A Mathematical Tools components. But since we only will write the indexed expressions in a Cartesian coordinate system, we have no need for such distinctions and can therefore keep it simple. First, the rules: 1. If a given letter index appears only once in each term of an equation, it is called a free index and the equation holds for all possible values of the index. If this index is ‘relabeled’ with a new letter, it must be relabeled consistently in every term. 2. If a letter index appears exactly twice in a term, then summation over all possible values of the index is implied. Such an index is called a dummy index. This index can be relabeled with a different letter (as long as the new letter does not conflict with another index), without affecting the other terms in the equation. 3. No letter index may appear more than twice in a given term. Some examples help to demonstrate the application of these rules: • The equation a = b + c can be represented in component form as ai = bi + ci. This obviously implies that the equation holds for all values of the free index i (1, 2, 3 in three-dimensional space, or just 1 and 2 in the plane). It would be a violation of rule 1 to change this to ai = bi + cj ; however, it is perfectly okay to change it to aj = bj + cj . • The dot product of two vectors, a · b, can be represented as ai bi, which implies a1b1 + a2b2 + a3b3 by rule 2 (when the dummy index i extends over the range 1, 2, 3). This can be relabeled as aj bj , provided j does not already appear as a free index or another dummy index in the term. • Note that multiplication of components is commutative, as it is for any scalar quantities, so we can reorder the components within a term however we find convenient. Thus, ai bi = biai. • By rule 3, the expression ai bici is meaningless. For example, the vector expression a = (b · c)d may be represented in component form as ai = bj cj di.Onemayalso relabel the dummy index j with another letter, such as k,toformai = bk ck di, but cannot relabel it with the letter i. Such relabeling can often be helpful for proving identities. For example, expression ai bicj dj fk − ambmcl dl fk vanishes, which is only obvious once the dummy indices in the second term are replaced as follows: m → i and l → j. The components of higher-rank tensors have the same number of indices as their rank; for example, Aij are the 9 (4 in two dimensions) components of a rank-two tensor, A. There are two tensors of special significance. The first is the Kronecker delta, defined as follows: 1, i = j δij ·= (A.1) 0, i j Note that the Kronecker delta is symmetric with respect to its indices, δij = δji.It has the property of ‘picking off’ a single component of a vector, e.g., δijvj = vi.To see this clearly, write out the sum completely, when, for example, i = 2: δ2j vj = δ21v1 + δ22v2 + δ23v3. (A.2) A.1 Some Vector Calculus Results 391 Only the middle term survives, and is equal to v2. This makes it clear that the Kronecker delta just describes the components of the identity tensor, 1. This tensor is isotropic, which means its components are the same in every Cartesian coordinate system. The cross (or vector) product between two vectors, denoted vectorially by c = a×b, uses another special isotropic tensor, this time of rank 3, called the permutation symbol, ijk. This symbol has the definition ⎧ ⎪⎨ 1, if ijk is an even permutation of 123 ijk ·= −1, if ijk is an odd permutation of 123 (A.3) ⎪ ⎩ 0, if any of ijk are equal Note that, for any three values of i, j and k, ijk = jki = kij = −ikj = −kji = −jik. By this definition, the cross-product operation can be denoted in index notation as follows: The ith component of c is ci = ijkaj bk . (A.4) The easiest way to see how this gives the usual relationship between components in the cross product is to write it out in full. Note that j and k are dummy indices in this expression. For example, when i = 1, c1 = 111a1b1 + 112a1b2 + 113a1b3 + 121a2b1 + 122a2b2 + 123a2b3 + 131a3b1 + 132a3b2 + 133a3b3. (A.5) By the definition of the permutation symbol, only the sixth and the eighth terms in this expression survive, and the result simplifies considerably to c1 = a2b3 − a3b2, as expected. A very useful relationship between the Kronecker delta and the permutation symbol is ijkklm = δilδjm − δimδjl. (A.6) This relationship holds in both two and three dimensions, and is generally used to derive identities involving two cross products: for example, a ×(b × c) = (a · c)b − (a · b)c. In two dimensions, it is often used with at least one of the three vectors or vector operators oriented in the out-of-plane direction. In such cases, one must take care to keep track of the range of each of the indices in (A.6). The reader is invited to use index notation to prove the following useful identity for two differentiable vector fields, a and b: ∇×(a × b) = ∇·(ba)−∇·(ab) = a(∇ · b)−b(∇ · a) + b ·∇a − a ·∇b. (A.7) 392 A Mathematical Tools A.1.2 Derivatives of Fundamental Solutions of Laplace’s Equation We can use index notation to develop some useful results concerning the derivatives of the distance r (or its inverse) from the origin to some point at x. First, note that we can write the square of this distance, in any dimension, as r2 = |x|2 = xk xk , where k is a dummy index. The most basic result we need in this context is ∂xi = δij. (A.8) ∂xj In other words, the derivative of a coordinate with respect to another coordinate is either 0 if their corresponding axes are different, or 1 if their axes are the same. Then, we can easily obtain the following: ∂r2 ∂ ∂xk = (xk xk ) = 2xk = 2xk δki = 2xi . (A.9) ∂xi ∂xi ∂xi But we can also use the product rule on this derivative, ∂r2 ∂r = 2r . (A.10) ∂xi ∂xi Combining these two results, we get the very useful identity ∂r xi = . (A.11) ∂xi r The right-hand side comprises the components of a unit vector from the origin toward the point. All of our other results in this discussion make use of (A.11) and the chain rule. For example, in two dimensional contexts, the fundamental solutions of Laplace’s equation in external regions are log r and its derivatives, so it is helpful to calculate a few of these: ∂ 1 ∂r xi (log r) = = , (A.12) ∂xi r ∂xi r2 followed by ∂2 ∂ xj δij 2xi xj (log r) = = − , (A.13) ∂xi ∂xj ∂xi r2 r2 r4 and ∂3 ∂ δjk 2xj xk 2xiδjk 2xj δki 2xk δij 8xi xj xk (log r) = − = − − − + . ∂xi ∂xj ∂xk ∂xi r2 r4 r4 r4 r4 r6 (A.14) Analogously, in three dimensions, 1/r and its derivatives form the fundamental solutions, so we calculate the first few of these: A.1 Some Vector Calculus Results 393 Fig. A.1 Schematic of a volume V and enclosing surface S ∂ 1 1 ∂r xi = − = − , (A.15) ∂xi r r2 ∂xi r3 and then, ∂2 1 ∂ xj δij 3xi xj = − = − + , (A.16) ∂xi ∂xj r ∂xi r3 r3 r5 and again, ∂3 1 ∂ δjk 3xj xk 3xiδjk 3xj δki 3xk δij 15xi xj xk = − + = + + − . ∂xi ∂xj ∂xk r ∂xi r3 r5 r5 r5 r5 r5 (A.17) A.1.3 The Divergence Theorem and Some Relevant Uses Consider a volume V and an enclosing surface S with outward unit normal n, as depicted in Fig. A.1. The divergence theorem allows us to relate integrals over V, involving some differentiable field quantity defined over that volume, to fluxes through the enclosing surface. This is an extremely powerful tool. The Generalized Divergence Theorem Though we are used to thinking of the divergence theorem applied to vector fields, in the most general form of the divergence theorem this field quantity is a tensor field of any rank.
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