Lecture Notes & Information

Syllabus (課程摘要)

Information sheet (課程資訊)

Lesson 01: Introduction to PDEs & modeling of 1-D wave equations

Lesson 02: Separation of variables & D'Alembert solutions

Lesson 03: Heat equation with different BCs

Lesson 04: Nonhomogeneous PDEs & BCs

Lesson 05: Solving PDEs by integral transforms

Lesson 06: 2-D wave equation in Cartesian and polar coordinates

Lesson 07: Laplace's equation

Midterm histogram (Mean=65.5, Standard deviation=19.0)

Lesson 09: Complex numbers & functions

Lesson 10: Complex integration

Lesson 11: Power series

Lesson 12: Taylor series & uniform convergence

Lesson 13: Laurent series & residue

Lesson 14: Integral by residue theorem

Lesson 15: 2-D potential theory by complex analysis

Final exam histogram (Mean=66.3, Standard deviation=14.4)

Semester grades histogram (Mean=74.4, Standard deviation=15.8)

Homework Library

HW1 Problem Set: due by 3/10

HW1 Solutions

HW2 Problem Set: due by 3/24

HW2 Solutions

HW3 Problem Set: due by 4/7

HW3 Solutions

HW4 Problem Set: due by 4/14

HW4 Solutions

HW5 Problem Set: due by 5/12

HW5 Solutions

HW6 Problem Set: due by 6/2

HW6 Solutions

Hall of Fame

Jean-le-Rond D'Alembert (1717-1783): French mathematician. In 1747, he applied the differential calculus to the problem of a vibrating string, and arrived at a partial differential equation u_tt = u_xx. He succeeded in showing that the solution is u = f(x+t)+y(x-t).

Jean Baptiste Joseph Fourier (1768-1830): Narrowly escaped from the guillotine during the French Revolution, and participated in Napoleon's expedition to Egypt, Fourier is most known by his great contribution to the theory of heat. In 1807, he completed a memoir "On the Propagation of Heat in Solid Bodies", in which he derived the differential equation of heat transfer and proposed expansions of functions as trigonometric series. His theory, however, failed to convince the authorities of the time, such as: Lagrange, Laplace, Poisson, and Biot. His essay was not published until 1822.

Pierre-Simon Laplace (1749-1827): French physicist and mathematician. His work "Celestial Mechanics" translated Newton's geometrical study of mechanics to the one based on calculus, known as physical mechanics. He borrowed Lagrange's concept of potential, but brought it to new heights (Laplace's equation). He used Laplace transform (originally discovered by Euler) in his work on  probability theory. He was the first to publish the value of Gaussian integral.

Hermann von Helmholtz (1821-1894): German scientist. Trained as a medical doctor, he made great contributions in physiology and physics. He published his idea of "conservation of energy" in 1847, arguing that the seeming energy loss is simply an energy transfer to heat. He solved equations of ideal fluid, showing the characteristics of vortices. This initiated the theory of ether, claiming that it's the only substance in the cosmos, and all physical phenomena could be accounted for in terms of its properties. However, his attempt to deduce all electromagnetic effects based on ether did not succeed by the time of his death, when the classical physics reached its limit.

Friedrich Wilhelm Bessel (1784-1846): German astronomer. Left formal education at age of 14, he wrote a recognized paper on Halley's comet in 1804 and began his first professional job in an observatory. In 1809, he was awarded doctorate on the recommendation of Gauss, such that he could take the offer as the director of Königsberg Observatory. In 1817, he introduced "Bessel functions" in the study of determining the motion of three bodies moving under mutual gravitation.

Adrien-Marie Legendre (1752-1833): French mathematician. He introduced "Legendre functions" to determine the attraction of an ellipsoid at any exterior point. His result was highly praised by Laplace, and resulted in his inclusion in the Academy of Science in Paris in 1783. He also contributed to elliptic functions and number theory. 

Johann Dirichlet (1805-1859): German mathematician. He studied in Paris since 1822, in contact with Fourier, Laplace, Legendre, and Poisson. In 1825, he presented his partial proof of Fermat's last theorem for n=5, which brought him instant fame. Recommended by Alexander von Humboldt, he returned to teach in Breslau (1827), Berlin (1828), and Göttingen (1855). The work of proving the stability of solar system led him to the Dirichlet problem concerning harmonic functions with given boundary conditions. He also rigorously proved the convergence conditions of Fourier series.

Augustin-Louis Cauchy (1789-1857): French mathematician. Switching his engineering career to mathematics in 1813, he held a position at the École Polytechnique in Paris until 1830, when he refused to take oaths of allegiance toward King Louis-Phillipe. He traveled to Turin (1831), Prague (1833), and managed to retake his École Polytechnique post in 1848. He was well known for his contribution to complex analysis, clarifying Calculus by limits and continuity, rigorously proving the Taylor's theorem. The Cauchy-Riemann equations were first seen in his 1814 paper.

Georg Friedrich Bernhard Riemann (1826-1866): German mathematician. He studied mathematics under Dirichlet at University of Berlin during 1847-1849. His PhD thesis (1851), supervised by Gauss at University of Göttingen, studied the theory of complex variables, introduced topological methods into it (Riemann surfaces). His lecture about geometry laid foundation of Einstein's general relativity. He was appointed as professor, then chair of Mathematics of Göttingen in 1857, and 1859, respectively.

Jacques Hadamard (1865-1963): French mathematician with Jewish background. He earned his doctorate in 1892 for a thesis first examining complex analytic functions and singularities. His work in 1893 on matrix determinant leads to the "Hadamard matrices", which are important in coding theory. In 1896, he proved the prime number theory: the number of primes<n grows as fast as n/(ln n), which was conjectured since the 18th century. He was devoted to the "Dreyfus case", opposing the injustice perpetrated against a man in the name of reason of state. He took refuge in the USA during the Nazi occupation of France (1940-1944) as a visiting scholar in the Columbia University.

Brook Taylor (1685-1731): English mathematician. He graduated from St. John's College of Cambridge in 1709.  His book of 1715 showed the ideas of integration by parts, and Taylor's series expansion, though other mathematicians (J. Gregory, Newton, Leibniz, J. Bernoulli, de Moivre) had discovered the same series independently.

Colin Maclaurin (1698-1746): Scottish mathematician. He graduated from the University of Glasgow in 1712 at the age of 14 with a thesis developing Newton's gravitation theory. He became professor of mathematics in the University of Aberdeen in 1717, and switched to the University of Edinburgh in 1725 under the recommendation of Newton. He published 2 volume "Treatise of Fluxions" in 1742 in an attempt to put Newton's calculus on a rigorous ground by geometrical methods. He also provided the integral test for the convergence of infinite series.