AMATH 401: Vector Calculus and Complex Variables

Course Description

Textbook

References

The mathematics library has a number of Springer ebooks that can be downloaded and used as textbooks for the complex variables portion of the course. For examples, please see:

Agarwal, R. P., Perera, K., and Pinelas, S. 2011. An Introduction to Complex Analysis. Springer, New York.
Link to Textbook

Cohen, H. 2007. Complex Analysis with Aplications in Science and Engineering. Springer, New York.
Link to Textbook

Ponnusamy, S. and Silverman, H. 2006. Complex Variables with Applications. Birkhauser, Boston.
Link to Textbook

Some references for the vector calculus section are:

Davis, H. F. and Snider, A. D. 1995. Introduction to Vector Analysis. Wm. C. Brown Publishers, Dubuque, IA.
Schwartz, M., Green, S. and Rutledge, W. A. 1960. Vector Analysis with Applications to Geometry and Physics, Harper & Brothers, New York, NY.
Young, E. C. 1993. Vector and Tensor Analysis. Marcel Dekker, New York, NY.

And some more references for complex analysis are:

Ablowitz, M. J. and Fokas, A. S. 1997. Complex Variables: Introduction and Applications. Cambridge University Press, Cambridge, UK.
Brown, J. W. and Churchill, R. V. 1996. Complex Variables & Applications. McGraw-Hill, New York, NY.
Kwok, Y. K. 2002. Applied Complex Variables for Scientists and Engineers. Cambridge University Press, Cambridge, UK.
Marsden, J. E. and Hoffman, M. J. 1999. Basic Complex Analysis. W. H. Freeman, New York, NY.
Mathews, J. H. and Howell, R. W. 2001. Complex Analysis for Mathematics and Engineering. Jones and Bartlett, Sudbury, MA.
Saff, E. B. and Snider, A. D. 2003. Fundamentals of Complex Analysis with Applications to Engineering and Science. Prentice-Hall, Upper Saddle River, New Jersey.
Spiegel, M. R. 1968. Schaum's Outline of Complex Variables. McGraw-Hill, New York, NY. .LP Zill, D. G. and Shanahan, P. D. 2009. A First Course in Complex Analysis with Applications. Jones and Bartlett Publishers, Sudbury, MA.

Syllabus

(1) Vector analysis (4 weeks)

Vector fields and vector calculus.
Orthogonal curvilinear coordinates.
Gradient, divergence, and curl.
Line, surface, and volume integrals.
Green's theorem, Stokes's theorem, and divergence theorem.

(2) Complex analysis (6 weeks)

Complex numbers.
Functions of a complex variable.
Analyticity.
Integration.
Cauchy's theorem and the Cauchy integral formula.
Taylor and Laurent series.
Calculus of residues and contour integration.
Fourier and Laplace transforms.

Learning objectives and instructor expectations

Schedule and Homework

Follow links in the table below to obtain a copy of the homework in latex (.tex) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. An item shown below in plain text is not yet available.

Class Summaries

class summaries

Grading

Tutorials

Matlab code illustrating cylindrical and spherical coordinates: cylsph.m

Matlab code to check the Cauchy integral formula and a Laurent series: believe.m