Math 321 - Applied Mathematical Analysis

• Prerequisites: Math 234.
• Frequency: Fall (I), Spring (II)
• Student Body: Students in the Physical Sciences and Engineering. Applied math majors.
• Credits: 3. (N-A)
• Recent Texts: Advanced Engineering Mathematics, Michael D. Greenberg, Prentice Hall; Mathematical Methods for Physicists, Arfken and Weber.
• Course Coordinator: Fabian Waleffe
• Background and Goals: This is the first semester in the standard sequence in applied advanced calculus at the math department. This sequence is aimed at applied math majors and students in the sciences and engineering who need knowledge of advanced calculus concepts and their applications beyond those of the standard calculus sequence. The first half of the course is dedicated to vector and tensor calculus and the second part is an introduction to complex analysis. The course emphasizes understanding of the geometrical concepts and covers many linear algebra concepts in the restricted context of three-dimensional spaces. This is a good first course in the study of complex analysis and its applications.
• Alternatives: a deeper, proof-oriented version of much of this material is covered in 521, 522, 561 and 623. These courses are recommended/required for math-majors
• Subsequent Courses: Math 322

Content coverage:

• Vectors and Vector Spaces
  • Bases, components, linear (in)dependence, dot product, orthonormal bases
  • Cross-product, Levi-Civita symbol, mixed ("triple-vector") products and determinants
  • Vector functions of a scalar variable
  • Applications to motion of a particle, a system of particles, a rigid body
  • Cartesian coordinates and orthogonal transformations
• Matrices and Tensors
  • Definition of a matrix and basic operations
  • Orthogonal matrices, Gram-Schmid algorithm and Euler angles
  • Eigenvalues and eigenvectors
  • Linear systems, geometric interpretations
  • Tensor product and Tensors, dyadic and indicial notation
  • Applications to continuum mechanics, projections, reflections, rotations
• Vector Calculus
  • grad, div, curl, and combinations, Laplacian
  • Line, surface and volume integrals
  • Divergence and Stokes's Theorem
  • Irrotational Fields
  • Applications to Electromagnetism and kinematics of continua.
  • Coordinate transformations, orthogonal coordinates, polar and spherical coordinates
  • Curvilinear coordinates, implicit function theorem
• Complex Variables
  • Functions of a Complex Variable
  • Conformal Mapping
  • Contour Integration
  • Taylor Series, Laurent Series and the Residue Theorem