Geometry 2010/11
Lecturer: Prof Salamon
Exam 27 June 2011
Exam 15 July 2011
Exam 15 June 2009
| Course synopsis Each module represents at least 12 hours (roughly 2 weeks' material) | |
|---|---|
| "Algebra" | "Analitica" |
|
Matrices and linear systems Matrices, sums and products Powers and inverses of square matrices Systems of equations, matrix form Elementary row operations Row reduction and rank Gauss(-Jordan) method |
Vector analysis Vectors in space, operations on them Scalar product, distances, angles Cross product, triple product, areas, volumes Plane geometry, lines, conics Equation of a plane in space Parametric equation of a line |
|
Rank and dimension Solving a general system Row equivalence Subspaces of Rn Bases of subspaces Dimension of subspaces Spaces generated by rows and columns |
Differential calculus Curves in space, tangent vectors Arclength, line integrals Functions with domain in Rn, limits Continuity, topological notions Partial derivates, the gradient Vector-valued functions, Jacobian matrix |
|
Linear mappings The notion of a vector space Bases, isomorphic vector spaces Linear mappings The matrix of a linear map Kernel and image, their dimensions The sum of two subspaces, Grassmann's formula |
Functions of two variables The graph of z=f(x,y), paraboloids Differentiability, tangent plane Second partial derivatives, Hessian matrix Taylor expansion of f(x,y) Critical points Extreme values |
|
Diagonalization Linear transformations, change of basis Eigenvalues, determinants Eigenvectors, eigenspaces, multiplicity Symmetric matrices, quadratic forms Orthogonal matrices and rotations Diagonalization of square matrices |
Surfaces and quadrics Surfaces of revolution, spheres Ellipsoids and hyperboloids Rotations in Rn Quadrics, planes and lines Surfaces defined implicitly Normal vector, tangent plane |
| Provisional lecture plan L = 60 hours and E = 39 hours | |||||
|---|---|---|---|---|---|
| 14/03 | L | Matrices, sums and products [Summary] | Notes, pp 1-4 | ||
| 15/03 | L | Powers and inverses of square matrices [Summary] | Notes, pp 5-8 | ||
| 17/03 |
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| 18/03 | E | ||||
| 18/03 | E | Vectors in space, operations on them | Notes, pp 33-34 | ||
| 21/03 | L | Systems of equations, matrix form [Summary] | Notes, pp 9-10 | ||
| 22/03 | L | Elementary row operations [Summary] | Notes, pp 13-14 | ||
| 24/03 | L | Gauss(-Jordan) method [Summary] | Notes, pp 17-18 | ||
| 25/03 | E | Notes, pp 10,12,16 | |||
| 25/03 | E | Scalar product, distances, angles |
Notes, pp 35-36 | ||
| 28/03 | L | Row reduction and rank [Summary] | Notes, p 18-19 | ||
| 29/03 | L | Solving a general system [Summary] | Notes, pp 21-22 | ||
| 31/03 | L | Row equivalence [Summary] | Notes, pp 11,15,20 | ||
| 01/04 | E | Notes, pp 22-23 | |||
| 01/04 | E | Cross product, triple product, areas, volumes | Notes, pp 39-40 | ||
| 04/04 | L | Subpsaces of Rn [Summary] | Notes, pp 25-26 | ||
| 05/04 | L | Bases of subspaces [Summary] | Notes, pp 29-30 | ||
| 07/04 | L | Dimension and rank [Summary] | Notes, pp 31-32 | ||
| 15/04 | E | ||||
| 15/04 | E | Spaces generated by rows and columns | Notes, pp 26-27 | ||
| 11/04 | L | The notion of a vector space [Summary] | Notes, pp 49-50 | ||
| 12/04 | L | More bases and subspaces [Summary] | Notes, pp 51-55 | ||
| 14/04 | L | Linear mappings [Summary] | |||
| 15/04 | E | ||||
| 15/04 | E | Notes, pp 50-51 | |||
| 18/04 | L | Bases and linear mappings [Summary] | Notes, pp 57-59 | ||
| 19/04 | L | Linear maps, matrices, systems [Summary] | |||
| EASTER | |||||
| 28/04 | L | Linear maps, kernels, images [Summary] | |||
| 02/05 | L | The sum of two subspaces [Summary] | Notes, pp 65-66 | ||
| 03/05 | L | Eigenvectors and eigenvalues [Summary] | Notes, pp 69-71 | ||
| 05/05 | L | Eigenspaces [Summary] | Notes, pp 77-78 | ||
| 06/05 | E | ||||
| 06/05 | E | Determinants | |||
| 09/05 | L | Diagonalization of square matrices [Summary] | Notes, pp 81-82 | ||
| 10/05 | L | Diagonalizing symmetric matrices [Summary] | Notes, pp 85-86 | ||
| 12/05 | L | Orthogonal matrices and quadratic forms [Summary] | |||
| 13/05 | E | ||||
| 13/05 | E | Plane geometry, conics | Notes, p 91 only | ||
| 16/05 | L | Equation of a plane [Summary] | Notes, pp 41-43 | ||
| 17/05 | L | Parametric equation of a line [Summary] | Notes, pp 45-46 | ||
| 19/05 | L | Curves in space [Summary] | MAI, pp279-283 | ||
| 20/05 | E | ||||
| 20/05 | E | ||||
| 23/05 | L | Line integrals along curves [Summary] | MAI, pp368-375 | ||
| 24/05 | L | Functions to and from Rn [SUMMARY] | MAI, p76, p285 | ||
| 26/05 | L | Partial derivatives, the gradient | MAI, p286 | ||
| 27/05 | E | ||||
| 27/05 | E | Graphs z=f(x,y), paraboloids | |||
| 30/05 | L | Chain rule for scalar functions [SUMMARY] | MAI, p288-289 | ||
| 31/05 | L | Scalar functions of 2 and 3 variables [SUMMARY] | MAII, p156-162 | ||
| 03/06 | E | ||||
| 03/06 | E | ||||
| 06/06 | L | Second order partial derivatives [SUMMARY] | MAII, p168-169 | ||
| 07/06 | L | First Taylor expansions [SUMMARY] | MAII p161 | ||
| 09/06 | L | Critical points [SUMMARY] | MAII, p174-180 | ||
| 09/06 | E | Extreme values (Room 1S at 4pm) | |||
| 10/06 | E | ||||
| 10/06 | E | ||||
| 13/06 | L | Conics and quadrics [SUMMARY] | Notes, p 93-95 | ||
| 14/06 | L | Quadrics, conics and lines [SUMMARY] | |||
| 16/06 | L | Examples and interpretations (at 1000) [Summary] | |||
| E | Model exam solutions (at 1130) | ||||
| 17/06 | E | ||||
| 17/06 | E | ||||
| 20/06 | L | Parametrization of surfaces [Summary] | |||
| 21/06 | L | Surfaces of revolution [Summary] | |||
| 23/06 | L | Tangents and normals (at 10.00) [Summary corrected] | |||
| E | Worked sample exercises | ||||
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References
Note: The summaries should suffice as a basis of theory for the exams !But more detail is available in the Notes and the following texts (not online).
MAI= Mathematical Analysis I by C. Canuto and A. Tabacco, Universitext, Springer
MAII= Mathematical Analysis II by C. Canuto and A. Tabacco, Universitext, Springer
Elementary Linear Algebra by Howard Anton, 10th edition, Wiley and Sons
Elementary Linear Algebra with supplemental applications by H. Anton and C. Rorres, 10th edition, Wiley, 2010