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Class: 3rd Year H/W UnitsSubject: Applied Mathematics Thr. 3 Lab. - 3×2 = 61.2.3.4.5.6.7.8.Complex numbers and hyperbolic functionsThe need for complex numbers, Manipulation of complex numbers, Polarrepresentation of complex numbers , de Moivre’s theorem, Complex logarithms andcomplex powers, Applications to differentiation and integration, Hyperbolic functionsSeries and limitsSeries, Summation of series, Convergence of infinite series, Operations with series,Power series,Convergence of power series; operations with power series, Taylor series, Evaluation oflimitsPartial differentiationDefinition of the partial derivative, The total differential and total derivative, Exact andinexact differentials , Useful theorems of partial differentiation, The chain rule, Changeof variables,Taylor’s theorem for many-variable functions, Stationary values of manyvariablefunctions, Stationary values under constraints, Envelopes, Thermodynamicrelations, Differentiation of integralsMultiple integralsDouble integrals, Triple integrals, Applications of multiple integrals, Change ofvariables in multiple integralsVector algebraScalars and vectors, Addition and subtraction of vectors, Multiplication by a scalar,Basis vectors and components, Magnitude of a vector, Multiplication of vectors,Equations of lines, planes and spheresUsing vectors to find distances, Reciprocal vectorsMatrices and vector spacesVector spaces, Basis vectors; inner product; some useful inequalities, Linear operators,Matrices, Basic matrix algebra, Functions of matrices, The transpose of a matrix, Thecomplex and Hermitian conjugates of a matrix, The trace of a matrix, The determinantof a matrix, Properties of determinants,The inverse of a matrix, The rank of a matrix, Special types of square matrix,Eigenvectors and eigenvalues Determination of eigenvalues and eigenvectors, Changeof basis and similarity transformations, Diagonalisation of matrices, Quadratic andHermitian forms, Stationary properties of the eigenvectors; quadratic surfaces,Simultaneous linear equationsVector calculusDifferentiation of vectors, Composite vector expressions; differential of a vector,Integration of vectors, Space curves, Vector functions of several arguments, Surfaces,,Scalar and vector fields, Vector operators,Gradient of a scalar field; divergence of avector field; curl of a vector field, Vector operator formulae,Cylindrical and sphericalpolar coLine, surface and volume integralsLine integrals, Evaluating line integrals; physical examples; line integrals with respectto ascalar, Connectivity of regions, Green’s theorem in a plane, Conservative fields andpotentials,Surface integrals, ,Evaluating surface integrals; vector areas of surfaces; physicalexamples