1 Curvilinear Coordinates

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1 MATHEMATICA PHYSICS PHYS-2106/3 Course Summary Gabor Kunstatter, University of Winnipeg April Curvilinear Coordinates 1. General curvilinear coordinates 3-D: given or conversely u i = u i (x, y, z), i = 1, 2, 3 (1) x = x(u 1, u 2, u 3 ), y = y(u 1, u 2, u 3 ), z = z(u 1, u 2, u 3 ) (2) calculate the following: (a) Infinitesmal displacements: d r = r i du i u i (b) Basis vectors r ê i = ui (3) r u i Be able to determine whether they are orthogonal (c) ine element ds 2 := d r d r (4) = (h 1 ) 2 (du 1 ) 2 + (h 2 ) 2 (du 2 ) 2 + (h 3 ) 2 (du 3 ) 2 Orthogonal only (d) Area element (orthogonal only) (e) Volume element (orthogonal only) d a 1 = h 2 h 3 ê 1, etc (6) dv u = h 1 h 2 h 3 du 1 du 2 du 3 (7) (5) 1

2 (f) Jacobian (general) J := = (u 1 u 2 u 3 ) (x, y, z) u 1 x u 2 x u 3 x u 1 y u 2 y u 3 y (8) u 1 z u 2 z (9) u 3 z dv u = Jdxdydz (10) 2. Specific examples are all derivable once you know ds 2. Specifically know: (a) Cylindrical (b) Spherical 2 Differential Operators 1. Geometrical/physical interpretation of A and A. Physical examples. 2. Know formulae (i.e. use geometrical intuition to construct): (a) gradient (b) divergence (c) laplacian 3. Know how to calculate (i.e. apply formulae) in general curvilinear orthogonal coordinates all four: grad, div, laplacian, curl 3 Conservative Forces 1. Know the four equivalent tests conservative forces: (a) F d r is path independent (b) F = U, for some potential energy function U (c) F = 0 2

3 (d) F d r = du is exact 2. Know how to prove equivalence 3. Know how to integrate exact differentials to get potential 4 ine, surface and volume integrals 1. Know how to arametrize curves and surfaces. 2. Calculate d r, d a for specific curves and surfaces. 3. Know how to choose coordinates and basis vectors wisely 4. Know your limits 5. Do the integrals 5 Divergence Theorem and Stokes Theorem 1. Know integral forms of grad, div, curl and laplacian 2. Divergence theorem 3. Stokes theorem (Green s theorem as special case) 4. Know geometrical interpretations of div and curl 5. Physical examples of div: (a) Continuity equation (b) Maxwell s equations (static): electric charge as source of electric field lines (Gauss law) 6. Physical examples of curl: (a) Maxwell s equations (static): E curl free (static), electric currcurrent J as source of curl of B 7. Use of Gauss law to derive electric field. 3

4 6 Fourier Series 1. Know qualitatively the Dirichlet conditions. To have a Fourier series a function must: (a) be periodic, with period, say; (b) be single-valued and continuous except for a finite number of finite discontinuities; (c) have finite number of maxima and minima within one period; (d) have a convergent integral of f(x) over one period. 2. Know how to calculate coefficients, using symmetry where possible f(x) = a [ ( ) ( )] 2πrx 2πrx a r cos + b r sin r=1 a r = 2 xo+ ( ) 2πrx dxf(x) cos x 0 b r = 2 xo+ ( ) 2πrx dxf(x) sin x 0 3. Understand periodic functions as a vector space (know proof) (11) (12) (13) 4. Know how to calculate Fourier series for non-periodic functions on an interval 5. Complex form of Fourier series 6. Parseval s theorem 7 Fourier Transform 1. Know how to calculate simple Fourier transform F (k) 2. Wave number k = 2π/λ and frequency ω = 2π/T 3. Reality condition: f(ω) = [ f( ω)] 4. Dirac Delta function: 4

5 (a) Defining property (b) Integral form: Fourier transform (c) Properties (d) Mathematical uncertainty principle 5. Quantum uncertainty principle (a) Mathematical uncertainty principle x k 1/2 (b) Features of quantum mechanics (qualitative only): i. state specified by complex normalizable wave function ψ(x, t) predicts only probabilities ii. measurements yield specific values of x (c) Calculation of < x >, < x > P(x)dx = ψ (x)ψ(x)dx (14) (d) Heisenberg uncertainty principle x p /2 is a directo consequence of de Broglie wavelength: λ = hc mv p = k (15) (e) momentum as a differential operator: ˆp = i d/dx so that using Parseval: < p >= dxψ (x)( i dψ/dx) = dk ψ (k) k ψ(k) (16) 6. Cosine and Sine Transforms: even and odd functions. 7. Convolution Theorem: know how to prove and to use. 8 aplace Transforms 1. Definition and how to calculate simple examples 2. Properties: 5

6 (a) inearity (b) Convolution (c) Differentiation, integration, scaling, translation, exponential multiplication 3. Use of Table 13.2 and properties to calculate inverse aplace transforms 9 Ordinary Differential Equations 9.1 General 1. Classification: order and degre 9.2 First degree, First Order dy = F (x, y) dx or A(x, y)dx + B(x, y)dy = 0 (17) 1. Separable 2. Exact: A = U/ x, B(x, y) = U/ y U(x, y) = c, solve for y(x) if possible. 3. Inexact: try integrating factor function of x only or y only. 4. Homogeneous: dy dx = A(x, y) B(x, y) Change variable to v = y/x separable in x, v = F (y/x) (18) 5. inear: dy + P (x)y = Q(x) (19) dx integrating factor µ(x) = P (x)dx, see above. 6. Bernouilli s equation (almost linear): dy dy + P (x)y dx dx = Q(x)yn (20) Change variables to v = y dy dx 1 n linear, see above 6

7 9.3 Higher degree, first order Define p := dy/dx. 1. Solvable for p = p(x, y) 2. solvable for x = x(y, p) 3. solvable for y = y(x, p 9.4 Higher Order General Considerations: 1. y = y c + y p 2. n linearly independent solutions in y c, Wronskian non-zero. 3. Boundary conditions fix integration constants 1. inear, Constant Coefficients (a) Try y c = e λx solve polynomial for λ (b) Find y p using educated guesses from f(x) and knowledge of y c. (c) aplace transform method 2. inear non-constant co-efficients, non-homogeneous (a) Exact: reduce to lower order (b) Variation of parameters (need to know y c ) (c) Green s function method (need to know y c ) 3. Second order non-constant coefficients, homogeneous series solutions 10 Partial Differential Equations 1. Recognize important physical PDEs: (a) Wave equation (b) Diffusion equation 7

8 (c) aplace s equation (d) Poisson s equation (e) Schrodinger equation (f) Maxwell s equation 2. Second order liner: (a) Classification: elliptic, hyperbolic, parabolic (b) General solution of 1-d wave equation 8