Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 321: Applied Mathematical Analysis I

Students in the Physical Sciences and Engineering. Applied Math, Engineering and Physics (AMEP) majors. 
 
Background and Goals: Math 321 covers vector algebra, vector calculus and an introduction to complex calculus. Math 321 was redesigned to provide mathematical preparation for Electromagnetism (PHYS 322) as well as further courses in Mathematics, Physics and Engineering such as Transport phenomena, Fluid and Solid Mechanics (ME 561, EMA 622) , Aerodynamics (EMA 521), Flight dynamics (EMA 523), Advanced dynamics (EMA 542), astrodynamics (EMA 550), satellite dynamics (EMA 642), Computer graphics, plasma physics (PHYS 525), differential geometry (MATH 561), for example. The course emphasizes understanding of the geometrical concepts and covers vector (Gibbs) notation, index notation (including summation convention) and linear algebra notation.  
 

Prior course: Math 320 (Diff eqs with linear algebra). Math 320, 321, 322 are a core sequence for mathematics, physics and engineering.

Subsequent Courses:  Physics 322 (E&M), Math 322 (PDEs, Fourier Series, Special functions)
 
Course Content: 
  • Vector Algebra
    • Cylindrical, Spherical and Cartesian representation of vectors. Points, coordinates and position vectors.
    • Bases, components, linear (in)dependence. Lines and planes.
    • Dot product, Kronecker delta, orthonormal bases
    • Cross product, Levi-Civita symbol
    • Index notation and summation convention.
    • Applications of dot and cross products. Mixed products and determinants. Rotation of vectors.
    • Rotation and reflection of bases. Orthogonal matrices, Euler angles.
    • (Optional) Basic matrix operations and concepts, linear systems of equations and geometrical interpretations.
  • Vector Calculus
    • Vector functions of a scalar variable
    • Applications to motion of a particle, a system of particles, a rigid body
    • Curves, Surfaces and Volumes and integrals over them. Examples from mechanics and electromagnetism.
    • Coordinate transformations, curvilinear coordinates, cylindrical and spherical coordinates. Jacobian
    • Grad, div, curl
    • Derivation of vector identities using vector and index notation. Laplacian.
    • Divergence and Stokes' Theorems
    • Irrotational fields. Solenoidal fields.
    • Applications to Electromagnetism and mechanics of continua.
  • Complex Variables
    • Complex algebra, series, radius of convergence
    • Functions of a Complex Variable, branch cuts
    • Conformal Mapping, Laplace's equation
    • Contour Integration and residue calculus
credits: 
3. (N-A)
semester: 
FallSpring
prereqs: 
MATH 376 or (MATH 234 and 319) (MATH 234 and 320), (MATH 234 and 340), (MATH 234 and 341) or (MATH 234 and 375) or graduate or professional standing

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, WI  53706

(608) 263-3054

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