Instructor Dragos Oprea
Lectures: MWF, 3pm-3:50pm, AP&M 5-402.
Office hours: Wednedays and Fridays from 2-3pm in AP&M 6-101. I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.
Course
Assistants		 Robert McGuigan
Course
Content		 This course is the third in the Mathematics Department's Honors sequence, Math 31H. Topics include: change of variables formula, integration of differential forms, exterior derivative, generalized Stoke's theorem, conservative vector fields, potentials.
Prerequisities: Math 31BH with a grade of B- or better.
Grade
Breakdown 		The grade is computed as the following weighed average:
  • Homework 20%, Midterm I 20%, Midterm II 20%, Final Exam 40%.
Textbook:J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Fourth Edition.
ReadingsReading the sections of the textbook corresponding to the assigned homework exercises is considered part of the homework assignment. You are responsible for material in the assigned reading whether or not it is discussed in the lecture. It will be expected that you read the assigned material in advance of each lecture.
Homework Homework problems will be assigned on the course homework page. There will be 8 problem sets, due certain Fridays at 4:30PM in the TA's mailbox. You may work together with your classmates on your homework and/or ask the TA (or myself) for help on assigned homework problems. However, the work you turn in must be your own. No late homework assignments will be accepted.
Midterm
Exams		There will be two midterm exams given on April 26 and May 24. There will be no makeup exams.
Final
Exam		 The final examination will be held on Wednesday, June 12, 3PM - 6PM. There is no make up final examination. It is your responsability to ensure that you do not have a schedule conflict during the final examination; you should not enroll in this class if you cannot sit for the final examination at its scheduled time.
Announcements and Dates
  • Monday, April 1: First lecture
  • Friday, April 26: Midterm I
  • Friday, May 24: Midterm II
  • Monday, May 27: Memorial Day. No class
  • Friday, June 7: Last Lecture
  • Wednesday, June 12: FINAL EXAM, 3-6PM pm
Exams
Lecture Summaries
  • Lecture 1: Fubini's theorem and iterated integrals. Changing the order of integration. Examples. The volume of the ball in n-dimensions.
  • Lecture 2: Determinants. Expansion along first column. Existence of determinants of multilinear, antisymmetric, normalized functions.
  • Lecture 3: Uniqueness of the determinant function. Column operations. Determinants of invertible matrices are nonzero. Determinants can be expanded along any column. Determinants of products are products of determinants.
  • Lecture 4: Permutations. Signature of permutations. Determinants via permutations. Row expansion and row operations. Eigenvalues revisited. Characteristic polynomials. Eigenvalues add up to the trace, and multiply to the determinant. Cayley-Hamilton theorem.
  • Lecture 5: Diagonalizable matrices via eigenvalue multiplicities. Similar matrices. Similar matrices have the same characteristic polynomials, eigenvalues, trace, determinant. Examples.
  • Lecture 6: Volumes and determinants. Linear transformations and volumes. Examples: volume of ellipsoids, area of ellipses. k-dimensional volumes of parallelograms in R^n.
  • Lecture 7: Linear change of variables and proof. Arbitrary change of coordinates and Jacobians. Examples. Polar coordinates.
  • Lecture 8: Cylindrical coordinates. Spherical coordinates. Volume in cylindrical and spherical coordinates. Examples.
  • Lecture 9: Parametrizations of manifolds. Graphs. Surfaces of revolution. Examples: paraboloids, torus.
  • Lecture 10: Volume of paremetrized manifolds. Examples: length, graphs, surfaces of revolution. Examples: surface area of paraboloid, surface area of the torus.
  • Lecture 11: More example of surface area. Area in spherical coordinates. k-forms in R^n as multilinear and alternating functions of k-vectors. The standard k-forms.
  • Lecture 12: All forms can be expressed in terms of the elementary forms. Wedge product and examples. Form fields (or differential forms).
  • Lecture 13: Integration of k-form fields over parametrized domains in R^n. Pullbacks of differential forms. Examples.
  • Lecture 14: Orientation of bases in R^n. Change of basis matrix. Examples.
  • Lecture 15: Orientation of manifolds. Generalizing the right hand rule by means of determinants. Normal vectors determine an orientation. The case of manifolds described by global equations. Orientation via non-vanishing top forms.
  • Lecture 16: Integration over oriented manifolds. Orientation preserving parametrizations. Examples.
  • Lecture 17: Orientation preserving diffeomeorphisms. Reparametrizations. Proof that the integral is independent of parametrization.
  • Lecture 18: Applications of forms to physics. Vector fields, work form, flux form, Faraday form.
  • Lecture 19: Examples of flux, flux in spherical coordinates. Flux, work, wedge product and cross product.
  • Lecture 20: Exterior derivative of forms. Exterior derivative and wedge products. Square of the exterior derivative. Divergence and curl. Connection between divergence, curl, work, flux.
  • Lecture 21: Divergence and curl of the gravitational field. Piece of a manifold, boundary, orientation of the boundary.
  • Lecture 22: Generalized Stokes' theorem. The fundamental theorem of calculus is a particular case. Proof for surfaces in R^n.
  • Lecture 23: Applications of Stokes' theorem. The fundamental theorem of calculus for line integrals. Gradient fields and path independence.
  • Lecture 24: Green's theorem for work and flux and connection with curl and div respectively. Path independence of work and flux means zero curl and div respectively. Potentials.
  • Lecture 25: Interpretation of curl and div. Gauss' theorem and examples. The classical Stokes theorem.
  • Lecture 26: Examples of Stokes' theorem. Review.