벡터해석 및 복소함수

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공학 >기계ㆍ금속 >기계공학
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2015년 2학기

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공학에 응용되는 기본적인 벡터 대수학 및 벡터 해석의 개념에 대한 이해와 복소 함수론의 기초에 대하여 강의한다.

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The algebra of vectors

차시별 강의

1.	The algebra of vectors	Vector algebra, Parallelogram law, Representation of vector using co-ordinates, Inner(Dot) product, Index notation, Summation convention, Kronecker's delta.
	The algebra of vectors	Equations of plane, Distance from a point to plane, Cross product.
2.	The algebra of vectors	Triple scalar product, Triple vector product, epsilon-delta identity
	The algebra of vectors	Examples of triple scalar and vector products, Spherical cosine law, Decomposition of vectors, Velocity of rotating body, Linear momemtum, Angular momemtum.
3.	Vector functions of a real variable	Position vector, Velocity vector, Acceleration vector, Plane vector in polar coordinates, Linear momemtum, Angular momemtum, Keper's laws, Curvature, Torsion, The Serret-Frenet equations of a curve.
	Vector calculus	Scalar and vector fields, Nabla, Gradient, Directional derivative, Steepest descent direction.
4.	Vector calculus	The Divergence, The Curl, Formulas involving Nabla.
	Vector calculus	Formulas involving Nabla continued, Solenoidal field, Line integral, Path independent, Potential.
5.	Vector calculus	Greens theorem in the plane, Simply connected region, Multiply connected region, Stokess theorem, Divergence theorem.
	Vector calculus	Greens theorem in the plane, Stokess theorem, Divergence theorem.
6.	Vector calculus	Examples of Stokess theorem, Divergence theorem, Mass conservation ( continuity equation), Charge conservation.
	Vector calculus	Generalized Divergence theorem, Generalized Stokess theorem.
7.	Vector calculus	Orthogonal Curvilinear Coordinates, Cylindrical Coordinates, Spherical Coordinates.
	Functions of a complex variable	Complex numbers, Complex plane, Polar form of complex numbers, Modulus, Argument, Complex variables, Conjugate of a complex number.
8.	Functions of a complex variable	Complex logarithmic fuction,Generalized power fuctions, Generalized exponential function,Branch point, Branch cut.
	Functions of a complex variable	Inverse circular and hyperbolic functions, Analytic functions of a complex variable, Cauchy-Riemann equations.
9.	Functions of a complex variable	The Cauchy-Riemann equations for orthogonal curvilinear coordinates, Line integrals of complex functions, Cauchys integral theorem.
	Functions of a complex variable	Cauchys integral formula
10.	Functions of a complex variable	Cauchys integral formula, Cauchy principal value
	Functions of a complex variable	Cauchys inequality, Liouvilles theorem, Fundamental theorem of algebra, Taylors series
11.	Functions of a complex variable	The Laurent series, Isolated singularities, Poles, Essential singularities.
	Functions of a complex variable	Gauss mean value theorem, Maximum modulus theorem, Minimum modulus theorem, Removable singularities, Residues, The argument theorem.
12.	Functions of a complex variable	Rouches theorem, Dirichlet problems, Poissons integral formula for a circle.
13.	Functions of a complex variable	Schwarz integral formula for a circle, Poissons integral formula for a half plane,Schwarzs integral formula for a half plane.
14.	Functions of a complex variable	The residue theorem, Evaluation of integrals.
15.	Functions of a complex variable	Jordans lemma, Integrals involving branch point.