Annexe 3
RECOMMENDED KNOWLEDGE IN
MATHEMATICS
1 - ALGEBRA
1.1 Set theory
Operations on ts, characteristic functions.
Maps, injectivity, surjectivity.
Direct and inver image of a t.
Integer numbers, finite ts, countability.
1.2 Numbers and usual structures
Composition laws; groups, rings, fields.
Equivalence relations, quotient structures.
Real numbers, complex numbers, complex exponential.
Application to plane geometry.
Polynomials, relations between the roots and the coefficients.
Elementary arithmetics (in Z/nZ).
1.3 Finite dimensional vector spaces (*)
Free families, generating families, bas, dimension.
Determinant of n vectors; characterization of bas.
Matrices, operations on matrices.
Determinant of a square matrix; expansion with respect to a line or to a column; rank, cofactors.
Linear maps, matrix associated to a linear map.
Endomorphisms, trace, determinant, rank.
Linear systems of equations.
1.4 Reduction of endomorphisms
Stable subspaces.
Eigenvalues, eigenvectors of an endomorphism or a square matrix; similar matrices;
geometrical interpretation.
Characteristic polynomial, Cayley-Hamilton theorem.
Reduction of endomorphisms in finite dimension; diagonalizable endomorphisms and matrices.
1.5 Euclidean spaces, Euclidean geometry
Scalar product; Cauchy-Schwarz inequality; norms and associated distances.
风油精的功效与作用
Euclidean spaces of finite dimension, orthonormal bas; orthogonal projections.
Orthogonal group O(E); orthogonal symmetries.
Orthogonal matrices; diagonalization of symmetric real matrices.
Properties of orthogonal endomorphisms of R² and R³.
(*) In veral countries linear algebra is studied only in R k or C k; the candidates from the countries are strongly advid to get familiar with the formalism of abstract vector spaces.
2 - ANALYSIS AND DIFFERENTIAL GEOMETRY
2.1 Topology in finite dimensional normed vector spaces珠宝鉴定中心
Open and clod ts, accumulation points, interior points.
Convergent quences in normed vector spaces; continuous mappings.
Compact spaces, images of compact ts by continuous mappings, existence of
extrema.
Equivalence of norms.
2.2 Real or complex valued functions defined on an interval
Derivative at a point, functions of class C k.
Mean value theorem, Taylor's formula.
Primitive of continuous functions.
Usual functions (exponential, logarithm, trigonometric functions, rational fractions).
Sequences and ries of functions, simple and uniform convergence.
2.3 Integration on a bounded interval
Integral of piecewi continuous functions.
Fundamental theorem of calculus (expressing the integral of a function in terms of a
primitive).
Integration by parts, change of variable, integrals depending on a parameter.
Continuity under the sign , differentiation under the sign .
Cauchy-Schwarz inequality.
2.4 Series of numbers, power ries
Series of real or complex numbers, simple and absolute convergence.
Integral comparison criterion, product of absolutely convergence ries.
Power ries, radius of convergence; function that can be expanded in a power ries on an interval.
Taylor ries expansion of e t, sin(t), cos(t), ln (1+t), (1+t)a where a is a real number.
2.5 Differential equations
Linear scalar equations of degree 1 or 2, fundamental systems of solutions.
Linear systems with constant coefficients.
meaty
Method of the variation of the constants.
Notions on non-linear differential equations.
2.6 Functions of veral real variables
Partial derivatives, differential of a function defined on R k.
Chain rule.
C1-functions; Schwarz theorem for C2-functions.
Diffeomorphisms, inver function theorem.
Critical points, local and global extrema.
Plane curves; tangent vector at a point, metric properties of plane curves (arc length,
curvature).
Surfaces in R³, tangent plane to a surface defined by a Cartesian equation F(x,y,z) = 0.
Physics Syllabus
The required knowledge syllabus for applicants who main examination subject is physics is detailed below. For applicants who condary examination subject is physics, only Newtonian mechanics of the material point as well as basic facts about Maxwell’s equations, including electrostatics, are required.
All applicants should know the numerical values of the basic constants of physics, as well as the orders of magnitude of the physical phenomena of nature.
The applicants should be able to display excellent standard mathematical skills.
I. MECHANICS
Newtonian mechanics
Mechanics of solids
西双版纳自治州Statics and mechanics of fluids
Applications of mechanics
车马冷着II. ELECTRIC CIRCUITS
III. ELECTRICITY AND MAGNETISM
Electrostatics
Magnetostatics
Electromagnetic waves
IV. OPTICS
Geometrical optics
Wave optics
V. THERMODYNAMICS
Perfect gas
First and cond principles of thermodynamics
Physical constants
The values of Planck, Boltzmann and Avogadro constants, the charge and the mass of the electron, the speed of light in vacuum as well as the electric permittivity and the magnetic permeability of the free space should be known to the applicants (at least two significant digits are required). For numerical applications, the SI unit system is recommended.
Orders of magnitude
Physics is an experimental science who purpo is to describe as quantitatively as possible the surrounding world. As a result, the applicants should be able to asss the orders of magnitude of the physical effects they are investigating.
The orders of magnitude of quantities such as the magnetic field of the Earth, the radius of the Earth, the acceleration of free fall at the surface of the Earth, the concentration of electrons in a typical metal, the wavelengths of the electromagnetic waves of the visible spectrum, the distance between two atoms in a solid or liquid, the Bohr radius of the fundamental state of the hydrogen atom, the size of the nucleus, are a minimum basis of required knowledge.
Minimal requirements of calculation skills
In order to be able to follow the teaching of natural sciences profitably, the applicants are suppod to master a number of calculation skills, generally taught during the first three years of university studies. The are:
Expansions
It is unlikely, when solving a physics problem, that one will not need to analyze the behaviour of a physical quantity A(x) in the neighbourhood of a particular value of the x variable. The usual expansions
有关学习的成语
are therefore a part of the required knowledge basis.
Derivatives and primitives of the functions of a single variable
For a function of the variable x, what is ?
Derivatives of the basic functions :
小组合作学习as well as f(g(x)).
Rules of derivation of the product and the quotient of two functions of a real variable. Primitives of the basic functions above.
Integration by parts,.
Convergence problems appearing when the integration interval is infinite are normally considered as a part of the mathematical examination. One should only know that, at infinity, the function should decrea faster than 1 / x while in the neighbourhood of x0 the function should not diverge faster than 1 / (x–x0).
Integral transforms (e.g., of Fourier, Laplace or Hilbert) are not a part of the entrance examination syllabus.
Functions of veral variables. Usual differential operators.
孩子哭What does mean ?
Calculation of the partial derivative with respect to an independent variable in the ca of a function of veral variables.
The expression of the usual operators (gradient, curl, divergence) is required in rectangular co-ordinates only.
Gradient of a function of the rectangular coordinates.
Let be the gradient operator where are unit vectors along the respective rectangular axes.
Compute . Converly, if is given, compute f .
Curl.
Let is a vector field. Compute .
If
Divergence.
Let is a vector field. Compute .
Laplacian and vector Laplacian.
Compute
Multiple integrals. Stokes, Gauss – Ostrogradski theorems.
Multiple integrals are often reducible in ordinary physics problems to simple integrals becau the integrants and surfaces (volumes) involved in the calculations prent symmetry properties (most often cylindrical or spherical symmetries). Apart from tho simple (yet important) cas, the explicit calculation of a multiple integral is not required.
Circulation of a vector field. What does mean ?
: the circulation of a vector field along a clod curve (contour) C is equal to t he flow of the curl of that field through the surface ∑ subtended by C.
: the flow of a vector field through a clod surface ∑ is equal to the integral of its divergence over the volume enclod by ∑.
An important corollary of the above theorem is the Gauss’ theorem, of particular ufulness to inver-square-law fields (namely, gravity and electrostatic ones).
Differential equations
Quite often in physics, one has to deal with first order, , or cond order, , differential equations.
No theorems concerning the existence of solutions and their regularity are required at the physics examination.
The solution (including the formal one) of a first order differential equation with parable variables should be known to the applicants.
