The course consists of two parts: vector analysis and field theory, complex variable function theory and integral transformation. The purpose of this course is to make students master the basic theories, concepts and methods of vector analysis and field theory, complex variable function theory and integral transformation, systematically cultivate and train students' ability to solve practical problems by using the ideas and methods of vector analysis and field theory, complex variable function theory and integral transformation, and lay a good mathematical foundation for subsequent professional courses and future practical work.
Vector analysis and field theory
Chapter 1 Analysis of Vector and Vector-valued Function Lecture 4
Geometric vector, geometric vector addition, number multiplication, product, cross product, mixed product and trident product of vectors, definition of vector-valued function, addition, product multiplication, compound, product operation, limit and continuity of vector-valued function, derivative of vector-valued function, volume fraction of vector-valued function, curve integral, surface integral, Gaussian formula and Stokes formula.
Chapter II Digital Field Hours: 2
Isosurface of quantity field, directional derivative of quantity field, concept of gradient, usage of Hamiltonian.
Chapter III Counting Hours: 6
Vector line of vector field, flux of vector field, divergence of vector field, circulation of vector field, density of circulation surface of vector field, curl of vector field, and the relationship between derivative of vector field function and divergence, curl and gradient of vector field.
Chapter 4 Three Special Forms of Vector Field 4 Hours
Conservation field, conservation field curl, conservation field potential function, tubular field, tubular field vector potential, harmonic field, harmonic function.
Complex variable function and integral transformation part
Chapter 1: Complex number and plane point set: 2 hours
Rectangular coordinate representation, triangle representation and exponential representation of complex numbers. Modulus and radiation angle of complex numbers, four operations of complex numbers. Plane region, neighborhood, aggregation point, closed set, isolated point, boundary point, boundary, connected set, region, simple connected region, multi-connected region.
Chapter 2: Analysis of function hours: 6
Concept of complex variable function, geometric representation of complex variable function. Limit and continuity of complex variable function, the concept of derivative and analysis of complex variable function, the conditions of complex variable function analysis, the definition and properties of complex variable elementary function (exponential function, logarithmic function, power function, trigonometric function).
Chapter 3: Integration hours of complex variable function: 6.
Definition and properties of complex variable function integral, Cauchy theorem, Cauchy theorem in complex connected region, Cauchy integral formula, infinite differentiability of analytic function.
Chapter 4: Series of class hours: 6
Complex series, the concepts of convergence, divergence and absolute convergence of complex series, the concept of convergence circle and the solution of convergence radius of power series, and the properties of power series in convergence circle. Platform expansion of analytic function, zero point of analytic function, order of zero point. Laurent expansion, the solution of analytical function Laurent expansion, and the classification of isolated singularities by Laurent expansion.
Chapter 5: Hours Remaining: 2
The concept of residue, the basic theorem of residue, the solution of residue, and the calculation of integral with residue.
Chapter 6: conformal mapping hours: 6
The geometric meaning of derivative, conformal mapping, fractional linear mapping, the properties of elementary conformal mapping, and the solution of conformal mapping between some simple regions.
Chapter 7: Laplace transform: 8 hours
Laplace transform, properties of Laplace transform, inverse Laplace transform, convolution, simple application of Laplace transform.
Chapter 8: Fourier transform hours: 8
Conditions of Fourier integration and convergence, Fourier transform, Fourier inverse transform, properties of Fourier transform, convolution, cross-correlation function, function and its Fourier transform, simple application of Fourier transform.