Fall 2020
- MATH 311-504: Topics in Applied Mathematics I
Time and venue: MWF 12:00–12:50 p.m., ZOOM meeting
Office hours (ZOOM meeting):
- MWF 11:00–11:45 a.m.
- TR 4:00–5:00 p.m.
- by appointment
Help sessions (ZOOM meeting):
- Monday–Thursday, 12:00-8:00 p.m.
Office hours during the finals (ZOOM meeting):
- Monday, November 30, 3:00–5:00 p.m.
- Tuesday, December 1, 3:00–5:00 p.m.
- Wednesday, December 2, 3:00–5:00 p.m.
- Thursday, December 3, 3:00–5:00 p.m.
- Friday, December 4, 3:00–5:00 p.m.
- Monday, December 7, 9:00 a.m.–2:00 p.m.
Quiz 1: Friday, August 28 (topic: systems of linear equations)
Quiz 2: Friday, September 4 (topics: matrix algebra, inverse matrix)
Quiz 3: Friday, September 11 (topic: determinants)
Test 1: Monday, September 21
Quiz 4: Friday, September 25 (topics: vector spaces, subspaces, span,
linear independence)
Quiz 5: Friday, October 2 (topics: basis, dimension, coordinates,
rank of a matrix)
Quiz 6: Friday, October 9 (topics: linear transformations, matrix of a
linear transformation)
Quiz 7: Friday, October 16 (topics: eigenvalues and eigenvectors,
diagonalization)
Test 2: Friday, October 23
Quiz 8: Friday, October 30 (topics: orthogonal complement,
orthogonal projection, least squares solutions)
Quiz 9: Friday, November 6 (topics: norms and inner products,
orthogonal sets, the Gram-Schmidt process)
Quiz 10: Friday, November 13 (topics: differentiation of vector-valued functions; gradient, divergence and curl)
Test 3: Friday, November 20
Quiz 11: Monday, November 23 (topics: multiple integrals, line integrals)
Quiz 12: Wednesday, November 25 (topics: area of a surface, surface integrals)
Final exam: Monday, December 7, 11:00 a.m.-1:30 p.m.
Course outline:
Part I: Elementary linear algebra
- Systems of linear equations
- Gaussian elimination, Gauss-Jordan reduction
- Matrices, matrix algebra
- Determinants
Leon/Colley: Chapters 1-2
Lecture 1: Systems of linear equations.
Lecture 2: Gaussian elimination.
Lecture 3: Row echelon form. Gauss-Jordan reduction.
Lecture 4: Gauss-Jordan reduction (continued). Applications of systems of linear equations.
Lecture 5: Matrix algebra.
Lecture 6: Diagonal matrices. Inverse matrix.
Lecture 7: Inverse matrix (continued).
Lecture 8: Transpose of a matrix. Determinants.
- Leon/Colley 1.3-1.4, 2.1-2.2
Lecture 9: Properties of determinants. Evaluation of determinants.
Part II: Abstract linear algebra
- Vector spaces
- Linear independence
- Basis and dimension
- Coordinates, change of basis
- Linear transformations
Leon/Colley: Chapters 3-4
Lecture 10: Vector spaces.
Lecture 11: Subspaces of vector spaces.
Lecture 12: Span. Spanning set.
Lecture 13: Linear independence.
Lecture 14: Review for Test 1. (additional review)
- Leon/Colley 1.1-1.5, 2.1-2.2, 3.1-3.3
Lecture 15: Basis and dimension.
Lecture 16: Basis and dimension (continued). Rank of a matrix.
Lecture 17: Nullity of a matrix. Basis and coordinates. Change of basis.
Lecture 18: Change of basis (continued). Linear transformations.
Lecture 19: Examples of linear transformations. Range and kernel. General linear equations.
Lecture 20: Matrix transformations. Matrix of a linear transformation.
Lecture 21: Matrix of a linear transformation (continued). Similar matrices.
Part III: Advanced linear algebra
- Eigenvalues and eigenvectors
- Diagonalization
- Orthogonality
- Inner products and norms
- The Gram-Schmidt orthogonalization process
Leon/Colley: Chapters 5-7
Lecture 22: Eigenvalues and eigenvectors. Characteristic equation.
Lecture 23: Eigenvalues and eigenvectors of a linear operator. Basis of eigenvectors.
Lecture 24: Diagonalization. Euclidean structure in Rn.
- Leon/Colley 5.1, 6.3, 7.3, 7.6
Lecture 25: Orthogonal complement. Orthogonal projection.
Lecture 26: Orthogonal projection (continued). Least squares problems.
Lecture 27: Review for Test 2.
- Leon/Colley 3.4-3.6, 4.1-4.3, 5.1-5.3, 6.1, 6.3
Lecture 28: Norms and inner products.
Lecture 29: Orthogonality in inner product spaces.
Lecture 30a: The Gram-Schmidt process.
Part IV: Vector analysis
- Main notions of vector analysis
- Review of multiple integrals
- Line and surface integrals
- Green's, Gauss' and Stokes' theorems
Leon/Colley: Chapters 8-11
Lecture 30b: Review of differential calculus.
Lecture 31: Differentiation in vector spaces.
- Leon/Colley 7.4, 8.1, 8.3
Lecture 32: Gradient, divergence, and curl. Review of integral calculus.
Lecture 33: Review of integral calculus (continued). Area and volume.
Lecture 34: Multiple integrals. Line integrals.
- Leon/Colley 7.7, 9.1-9.5, 10.1
Lecture 35: Line integrals. Green's theorem.
- Leon/Colley 8.2, 10.1-10.2
Lecture 36: Conservative vector fields. Area of a surface.
Lecture 37: Surface integrals. Gauss' theorem. Stokes' theorem.
Lecture 38: Review for Test 3.
- Leon/Colley 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3
Lecture 39: Review for the final exam.
- Leon/Colley 1.1-1.5, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.6, 6.1, 6.3, 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3
Lecture 40: Review for the final exam (continued).
- Leon/Colley 1.1-1.5, 2.1-2.2, 3.1-3.6, 4.1-4.3, 5.1-5.6, 6.1, 6.3, 8.1-8.4, 9.1-9.5, 10.1-10.3, 11.1-11.3