Duration:
Number of lectures per week: 3
Assessment:
End-of-year Examination: 3 hour paper in June
Description:
General introduction to vectors and linear vector spaces, vectors in
3-dimensions, application to 3-dimensional geometrical problems.
Euclidean spaces, vector addition (triangle
and parallelogram law), multiplication of a vector by a scalar, and the
properties of these two operations which make the set of all vectors
a linear vector space (over the reals).
The set of all real n-tuples, as an example of a linear vector space.
Resolution of vectors along any two non-zero non-parallel vectors
in dimension 2 and along any three non-zero non-coplanar vectors
in dimension 3. The notion
of linearly independent vectors and vector bases. Orthonormal bases,
scalar and vector products, triple scalar and triple vector products and
geometrical interpretations.
Rotation of an orthonormal basis and the relationship between the old and
the new components of a vector.
A new definition of a (cartesian) vector
based on this relationship. The Einstein summation and range conventions.
The Kronecker delta and the Levi-Civita symbol and applications to vectors.
Generalisation to vectors in dimension n.
Matrices; motivation and definition. Algebra of matrices and multiplication
of two matrices. Determinants of square matrices, motivation, definition
and main properties. Elementary row and column operations.
Gaussian elimination algorithm. Cofactor expansion of determinants.
Invertibility of matrices and the formula for the inverse.
Solution of a system of linear equations, Cramer's rule.
Eigenvalues and eigenvectors of matrices, diagonalization of matrices.
Application to linear ordinary differential equations
Review of calculus in 1-dimension, introduction to
partial differentiation, gradient operator and its
geometrical significance.
Taylor polynomials, Taylor series.
Maxima and minima (extreme values), local and absolute.
Critical points, 1st and 2nd derivative tests.
Extreme values subject to constraints, Lagrange multipliers.
Multiple and iterate integrals, line, surface and volume integrals, change of
variable, Jacobians.
References
G. H. Thomas Jr and R. L. Finney : Calculus and Analytic Geometry
D. E. Bourne and P. C. Kendall : Vector Analysis and Cartesian Tensors
B. Kolman : Introductory linear algebra with applications.
M. O'Nan : Linear algebra
L. W. Mansfield : Linear algebra with geometric applications
S. Lang : Undergraduate Analysis; Calculus of several variables
W. Rudin : Principles of mathematical analysis
Apr 27, 2003