Advanced mathematical topics (2020-2021)
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Contents: the course mainly provides an introduction to multilinear algebra (and differential geometry)
Prerequisites of Linear Algebra: Vector spaces and bases; Inner products. Dual vector spaces
and dual bases. Vectors and covectors and their transformations under change of basis.
Multilinear algebra: Tensors: general definition; covariant and contravariant tensors.
Symmetric and antisymmetric tensors. Pseudo-tensors: the Levi-Civita tensor.
Tensor transformations, contractions. The metric tensor; raising and lowering indices;
the Riesz representation theorem. Tensors examples: strain tensor.
Cartesian and Curvilinear coordinates (tangent space and cotangent space; differential
and gradient of a function). The Covariant derivative. The Christoffel's symbols. Divergence
and Laplacian in curvilinear coordinates.
Some suggested material for consultation
Books:
B.A. Dubrovin, A.T. Fomenko, S.P. Novikov
"Modern Geometry - methods and applications. Part I: the geometry of surfaces, transformation
groups, and fields" (Springer, 1984).
L. Florack, "Tensor calculus and differential geometry (course notes 2020)"
W-H Steeb, "Hilbert spaces, wavelets, generalised functions and modern quantum
mechanics"
P. R. Halmos, "Finite-Dimensional vector spaces" (Dover ed).
Dates for exams:
First exam: 2.2.2021 at 10 a.m.
Exercises (pdf), Sol. Exer. (pdf)
Second exam: xx.xx.2021 at 10 a.m.
Exercises (pdf), Sol. Exer. (pdf)
Third exam: XX.09.2021 at 10 a.m.
Exercises (pdf), Sol. Exer. (pdf)
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