Ingegneria informatica (Ακαδημαϊκό έτος 2019/2020) - Information and communication technologies engineering (riservato agli studenti della Helwan University, Cairo, Egitto)

Calculus 2

ν. μαθήματος 1: Sequences Outline Numerical Sequences Convergence of sequences		Michael Lambrou
ν. μαθήματος 2: Series		Michael Lambrou
ν. μαθήματος 3: Criteria for series convergence Outline Criteria for series convergence Ratio and root test		Michael Lambrou
ν. μαθήματος 4: Sequences and series of functions Outline Sequences of functions Derivatives and integrals of limits Series of functions		Michael Lambrou
ν. μαθήματος 5: Power Series Outline Power Series Radius of convergence Differentiation and integration of power series		Michael Lambrou
ν. μαθήματος 6: Taylor series Outline Maclaurin series Polynomial approximation Taylor expansion of functions		Michael Lambrou
ν. μαθήματος 7: Fourier series Outline Trigonometric expansion of functions Orthogonal functions Behaviour at discontinuties		Michael Lambrou
ν. μαθήματος 8: Functions of two variables Outline Functions of two variables Graph		Michael Lambrou
ν. μαθήματος 9: Continuity and Partial derivatives Outline Limit of functions of two variables Continuity Partial derivatives Higher - order partial derivatives		Michael Lambrou
ν. μαθήματος 10: Differentiability Outline Differentiability of functions Chain rules Directional derivatives		Michael Lambrou
ν. μαθήματος 11: Functions of three or more variables Outline Functions of many variables Partial derivatives Chain Rules Cylindrical and spherical transformations		Michael Lambrou
ν. μαθήματος 12: Extreme of functions Outline Maxima, minima, saddle points Criteria for extrema Extrema on the boundary		Michael Lambrou
ν. μαθήματος 13: Lagrange Multipliere Outline Extrema under constraints Lagrange multipliers		Michael Lambrou
ν. μαθήματος 14: Double Integrals Outline Volume approximation by rectangles Double integrals Integration		Michael Lambrou
ν. μαθήματος 15: Double integrals over regions Outline Double integrals over regions Iteration of integrals Techniques of integration		Michael Lambrou
ν. μαθήματος 16: Change of variables Outline Change of variables into polars Change of variable formula for polars General change of variables Jacobians		Michael Lambrou
ν. μαθήματος 17: Triple Integrals Outline Triple integrals over a rectangular domain Triple integrals over general domains Volumes as triple integrals		Michael Lambrou
ν. μαθήματος 18: Evaluation of triple integrals Outline Techniques of evaluating triple integrals Iteration of integrals		Michael Lambrou
ν. μαθήματος 19: Applications of integration Outline Volume of solids Mass of solids Center of mass Moment fo inertia Pappus Theorem		Michael Lambrou
ν. μαθήματος 20: Differential equations Outline Idea of differential equations Separation of variables Homogeneous equations		Michael Lambrou
ν. μαθήματος 21: First order differential equations Outline First order linear differential equations Integrating factor Exact equations		Michael Lambrou
ν. μαθήματος 22: Second order linear differential equations Outline Second order differential equations Homogeneous equations Application		Michael Lambrou
ν. μαθήματος 23: Second order inhomogeneous differential equations Outline Second order linear inhomogeneous differential equations Particular solutions Undetermined coefficients		Michael Lambrou
ν. μαθήματος 24: Higher order differential equations Outline Higher order linear differential equations Homogeneous case Inhomogeneous case Methods of variation of parameters		Michael Lambrou
ν. μαθήματος 25: Systems of differential equations Outline Linear systems of differential equations Method of D operators		Michael Lambrou
ν. μαθήματος 26: Course overview Contents Complex function theory Working with functions and integrals A transform to the rescue Using complex numbers		Simon Salamon
ν. μαθήματος 27: Using complex number Contents Representing complex numbers Exponents and conjugates Applications to roots and powers Functions of a complex variable		Simon Salamon
ν. μαθήματος 28: Holomorphic functions Contents Mappings from the complex plane to itself Limits of a complex variable Complex derivatives Polynomial functions		Simon Salamon
ν. μαθήματος 29: The Cauchy Riemann equations Contents Partial derivatives Recognizing holomorphic functions Vector calculus Harmonic functions		Simon Salamon
ν. μαθήματος 30: Power series Contents Polynomials and series Absolute convergence Radius of convergence Analytic functions		Simon Salamon
ν. μαθήματος 31: Contour integration Contents Paths and curves The velocity integral Integrals along curves Length estimates		Simon Salamon
ν. μαθήματος 32: Cauchy's theorem Contents Fundamental theorem of calculus Regions in the plane Existence of primitives Integrating a holomorphic function		Simon Salamon
ν. μαθήματος 33: Cauchy's integral formula Contents Existence of primitives A singular integrand Integrating around circles Deforming a contour		Simon Salamon
ν. μαθήματος 34: Laurent series Contents Taylor series Derivatives of holomorphic functions Series of reciprocals Convergence in an annulus		Simon Salamon
ν. μαθήματος 35: Residues and boundaries Contents Laurent coefficients Simple closed contours Cauchy’s Theorem for boundaries Computing integrals using residues		Simon Salamon
ν. μαθήματος 36: Singularities and integrals Contents Isolated singularities Poles and zeros Computing and using residues Trigonometric integrals		Simon Salamon
ν. μαθήματος 37: Polynomials and definite integrals Contents Fundamental theorem of algebra Rational functions More integrals involving roots of unity More trigonometric integrals		Simon Salamon
ν. μαθήματος 38: Further integration tecnique Contents Indented contours Semicircular estimates Logarithmic integrals Summation of series		Simon Salamon
ν. μαθήματος 39: Laplace transforms Contents Basic properties Further examples Transforms of derivatives Solving an initial value problem		Simon Salamon
ν. μαθήματος 40: Transforms calculus Contents Limiting values and integrals New transforms from old Some special integrals Applications to finding inverse transforms		Simon Salamon
ν. μαθήματος 41: The inverse Laplace transforms Contents Inverting rational functions Known examples of inverse transforms Contour integral interpretation Applying the inversion theorem		Simon Salamon
ν. μαθήματος 42: The theory of distributions Contents The set of test functions Linear functionals Distributions as limits Derivatives of distributions		Simon Salamon
ν. μαθήματος 43: Working with distributions Contents Operations on distributions Differentiation and limits Principal value integrals Infinite sums and series		Simon Salamon
ν. μαθήματος 44: Convolution of function Contents Motivating examples Convolution as a product Spaces of integrable functions Convolution with a distribution		Simon Salamon
ν. μαθήματος 45: The Fourier transform Contents First examples Spectral analysis Derivatives and products Transform of a convolution		Simon Salamon
ν. μαθήματος 46: Fourier inversion Contents The inversion theorem Proof of inversion Using a convolution The Schwartz space		Simon Salamon
ν. μαθήματος 47: Fourier transforms of distributions Contents Convergence in Schwartz space Tempered distributions The generalized Fourier transform Generalized inversion		Simon Salamon
ν. μαθήματος 48: Back to Laplace transforms Contents Laplace versus Fourier Laplace convolution An application to beam bending Laplace inversion		Simon Salamon
ν. μαθήματος 49: Derivatives, series and integrals Contents Another differential equation Solution by series Laurent coefficients Laplace transform of an integral		Simon Salamon
ν. μαθήματος 50: A final application Contents A partial differential equation The heat kernel Final example Acknowledgements		Simon Salamon

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Ingegneria informatica (Ακαδημαϊκό έτος 2019/2020) - Information and communication technologies engineering (riservato agli studenti della Helwan University, Cairo, Egitto)

Calculus 2

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Ingegneria informatica (Ακαδημαϊκό έτος 2019/2020) - Information and communication technologies engineering (riservato agli studenti della Helwan University, Cairo, Egitto)

Calculus 2

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