PHY425B, Summer '99

A. CONTENTS

1. Vector Algebra: a Review

2. Differential Vector Calculus

3. Integral Calculus for Vectors

4. Orthogonal Curvilinear Coordinates

5. Fourier Analysis - Series: Part I

6. Fourier Analysis - Series: Part II

7. Fourier Integrals: Part 1

8. Fourier Integrals: Part II

9. Boundary Conditions: Vibrating Strings, Heat Diffusion

10. Analytic Functions

11. Cauchy Integral Theorem, Laurent Series and Residue

12. Conformal Mapping

13. Cauchy Residue Theorem and Definite Integrals

B. THE SKILLS

Vector Algebra: a Review

Knowledge

• Write the definition or explain each of the following terms or concepts in one or two sentences: vector, scalar, equality of vectors, vector addition and subtraction, null vector, unit vector, scalar product, vector function, rectangular unit vectors i, j, k, components of vector, basis and span, direction cosines of coordinate transformation, orthogonality condition for direction cosines, a_ij, vector product.
• Write the definition of a vector field and scalar field and test whether a given vector or scalar satisfies the definition by considering of rotation of the coordinates.

Rule Application

• Compute scalar and vector products given two vectors with numerical values for their components.
• Compute scalar and vector triple products given three vectors with numerical values for their components.
• Prove simple vector identities starting with the laws of vector algebra, scalar products, and vector products.

Differential Vector Calculus

Knowledge

• Write down the definition or explanation of each of the following terms and concepts: del operator ▼, gradient (of a scalar field), geometric interpretation of gradient, divergence of a vector field, curl of a vector field, geometric meaning of divergence, Laplacian of scalar field, Laplace's equation ▼²φ = 0.
• Verify simple vector identities involving the del operator, including successive applications of del.

Rule Application

• Calculate the gradient and Laplacian of a scalar field given the analytic form of the scalar field.
• Calculate the divergence and curl of a vector field given the analytic form of the vector field.

Integral Calculus for Vectors

Knowledge

• Write down Gauss's divergence theorem.
• Write down Stokes's theorem.
• Recognize some of the alternate form of Gauss's theorem (such as Green's theorem).
• Recognize some of the alternate forms of Stokes's theorem.
• Prove the various forms of Gauss's theorem, starting with: ∫_S V · dσ = ∫_V ▼ · Vdτ.
• Prove the various forms of Stokes's theorem, beginning with: ∫_loop V · dλ = ∫_s (▼ × V) · dσ.

Rule Application

• Evaluate integrals involving derivatives of scalar and vector fields using the various integral theorems.

Orthogonal Curvilinear Coordinates

Knowledge

• Define or explain the terms and concepts as follows: coordinate transformation, transformation equations, curvilinear coordinates, scale factor, orthogonal coordinates, differential elements of arc length in curvilinear coordinates, Jacobian of a coordinate transformation, spherical polar coordinates (r,θ,φ), differential element of volume in curvilinear coordinates.
• Write down from memory the transformation equations between rectangular coordinates (x,y,z) and each of the following: spherical polar coordinates (r,θ,φ) and circular cylindrical coordinates (ρ,φ,z).

Rule Application

• Compute scale factors, unit vectors, arc lengths, surface and volume elements for each of the curvilinear coordinate systems in K2.
• Compute the gradient, divergence, curl, and laplacian in each of the curvilinear coordinate systems in K2.

Fourier Analysis - Series: Part I

Knowledge

• Write a definition or explanation of the following terms: periodic function, period, Fourier series or expansion, Fourier coefficients, Dirichet conditions, odd and even functions, half-range Fourier sine and cosine series.

Rule Application

• Graph periodic functions when given the algebraic form of the function over one period and the period. Also, test whether a function is even or odd, given an algebraic or graphical statement of the function.
• Compute Fourier coefficients and write down the appropriate Fourier series when given the function.
• Test whether a function satisfies Dirichet conditions. Write the limit of the appropriate Fourier series at a point, given that the function satisfied Dirichet conditions.

Fourier Analysis - Series: Part II

Knowledge

• Write a definition or explain the following terms: Parseval's identity for Fourier series, set of orthogonal functions, normalized functions, normalized coefficients, set of orthonormal functions, Kronecker delta function, orthonormal series, density or weight function, complete set of functions.

Rule Application

• Write down Fourier series in complex notation. Compute Fourier coefficients c_n in complex notation.
• Use Parseval's identity and Fourier series to evaluate limits of series numerically.
• Test whether a given set of functions is orthogonal and compute normalization coefficients for each function.

Fourier Integrals: Part 1

Knowledge

• Define or explain each of the following terms or concepts: Fourier integral expansion f(x), Fourier transform of f(x), Fourier cosine transform and inverse cosine transform, inverse Fourier transform of F(ω), Fourier sine transform and inverse sine transform.
• Write down from memory Fourier's integral theorem.

Rule Application

• Calculate Fourier transforms and inverse transforms when given the appropriate functions.
• Evaluate infinite integrals using Fourier integrals.
• Solve certain integral equations using Fourier transforms.

Fourier Integrals: Part II

Knowledge

• Define or explain each of the terms and concepts as follows: Parseval's identities for Fourier transforms, convolution of f(x) and g(x), convolution theorem for Fourier transforms, Dirac delta function.

Rule Application

• Evaluate infinite integrals using Parseval's identity for Fourier transforms.
• Solve integral equations using the convolution theorem.
• Solve problems involving the Dirac delta function.

Boundary Conditions: Vibrating Strings, Heat Diffusion

Knowledge

• Vocabulary: partial differential equation, order of P.D.E., solution of P.D.E., boundary conditions, boundary value problem, linear and nonlinear P.D.E., elliptic P.D.E., hyperbolic P.D.E., parabolic P.D.E., diffusion and heat conducting equation, vibrating string equation, Laplace's equation, Poisson's equation, Dirichlet boundary conditions, Neumann boundary conditions, Cauchy boundary conditions.
• Classify a given P.D.E. according to these characteristics: linear or nonlinear, homogeneous or nonhomogeneous, order, elliptic, hyperbolic or parabolic.
• Derive the diffusion equation.

Rule Application

• Write down the mathematical B.V.P. corresponding to a physical problem when given the problem in words.
• Compute the solution to a physical problem involving vibrating strings with ends clamped or temperatures in a slab or bar.

Analytic Functions

Knowledge

• Write down from memory the definition or explanation of the following: polar representation of a complex number, modulus or magnitude of a complex number, phase of a complex number, complex function of a complex variable (C.V.), real part of a complex function, imaginary part of a complex function, complex conjugate, DeMoiure's theorem, complex plane, logarithm function, branch of logarithm, principle branch, branch point, cut line or branch line, derivative of a function of C.V., Cauchy-Riemann (C.R.) conditions, analytic function, harmonic functions, exponential, hyperbolic and trigonometric functions.

Rule Application

• Perform simple calculations involving the elementary functions [i.e., sin(z), cos(z), sinh(z), cosh(z), exp(z).]
• Test whether a function is analytic by using the Cauchy-Riemann conditions.
• Construct the real (imaginary) part of an analytic function by using the C.R. conditions when given the imaginary (real) part of the function.

Cauchy Integral Theorem, Laurent Series and Residue

Knowledge

• Vocabulary: contour integral, Taylor series, Laurent senses, isolated singularity, pole of order n, simple pole, removable singularity, essential singularity, residue, double pole, simply connected region or domain, multiply connected region or domain.
• Write, and explain with a diagram, Cauchy's integral theorem for simply and multiply connected regions.
• Prove that ∫_loop z^m-n-1 dz = 2 π i δ_mn when the contour encircles the origin and is counterclockwise.

Rule Application

• Apply Cauchy's integral theorem and Cauchy's integral formulas to evaluate integrals in the complex plane.
• Determine Taylor's and Laurent's series for a given function: (a) using the Cauchy integral formulas; (b) using known expansions for common functions.
• Classify the singularities of a given function according to whether they are isolated singularities, poles (give orders n), branch points, removable singularities, or essential singularities. Calculate the residues at each pole.

Conformal Mapping

Knowledge

• Write the definition or explanation of each of the following terms and concepts: mapping and transformation, types of transformations: translation, rotation, stretching, inversion, linear, bilinear of fractional, conformal mapping, area magnification factor, linear magnification factor.

Rule Application

• Determine the region in the w-plane into which a given region in the z-plane is mapped by an analytic function f(z).
• Solve the boundary value problem involving Laplace's equation and Dirichet or Neumann B.C. by using conformal mapping techniques.

Cauchy Residue Theorem and Definite Integrals

Knowledge

• Write the definition or explain the meaning of the Cauchy principle value of an integral.
• Write down the Cauchy residue theorem when asked, and include the proper conditions on the functions involved.

Rule Application

• Evaluate various definite integrals using the Cauchy residue theorem.