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12695 Extension of Mathematical Methods - Five-year degree in Physics

Center

Faculty of Physics

Departament

Theoretical Physics

Lecturers in charge

Sin datos cargados

Met. Docent

METHODOLOGY
An exam with two sittings. Students' participation in practical lessons will be assessed and will constitute a part of the final mark.

Met. Avaluació

- -

Bibliografia

1.M.P.do Carmo, Differential Geometry of Curves and Surfaces, New Jersey, 1976
2.B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry-Methods and Applications II, Springer-Verlag,
1984
3.S. Weinberg, Gravitation and Cosmology, New York, 1972
4.R.M. Wald, General Relativity, Chicago, 1984

Continguts

CONTENTS
To introduce the basic concepts of differential geometry, emphasizing its applications to Physics.
THEORY PROGRAMME :
1) Theory of curves and surfaces in R3
Lesson I. Theory of curves.
1. Curves dependent on parameters in R3
2. Unitary, normal and binormal tangent.
3. Curvature and twist
4. Local canonical form

Lesson II. Regular surfaces in R3
1. Regular surfaces in R3
2. Tangent plane and normal vector
3. Differential of a function defined on a surface

Lesson III. Geometry of the Gauss Application
1. First fundamental form
2. Gauss application
3. Second fundamental form
4. Geometry of Gauss application. Gauss curvature and average curvature.
5. Weingarten equations.

Lesson IV. Intrinsic Geometry of Surfaces
1. Isometry and conformal applications
2. Christoffel symbols
3. Covariate derivative. Parallel transport.
4. Geodesics
5 Varieties and calculation in varieties

Lesson V. Varieties
1. Differentiable n-dimensional variety
2. Tangent space in a point of a variety

Lesson VI . Tensorial Algebra
1. Tensor in the space tangent to a variety
2. Tensorial product
3. Law of tensor transformation
4. Tensorial contraction
5. Tensor symmetrization and anti-symmetrization
6. External product of non mixed tensors.

Lesson VII. Differe