Real numbers, inequalities and intervals Function, domain and range Basic coordinate geometry Polar coordinates Mathematical induction Binomial theorem Combination of functions Symmetry in functions and graphs Inverse functions Complex numbers; real and imaginary forms Geometry of complex analysis Modulus-argument form of a complex number Roots of complex numbers Limits One-sided limits Derivatives Leibniz's formula Differentials Differentiation of inverse trigonometric functions Implicit differentiation Parametrically defined curves and parametric differentiation The exponential function The logarithmic function Hyperbolic functions Inverse hyperbolic functions Properties and applications of differentialability Functions of two variables Limits and continuity of functions of two real variables Partial differentiation The total differential The chain rule Change of variable in partial differentiation Antidifferentiation (integration) Integration by substitution Some useful standard forms Integration by parts Partial fractions and integration of rational functions The definite integral The fundamental theorem of integral calculus and the evaluation of definite integrals Improper integrals Numerical integration Geometrical applications of definite integrals Centre of mass of a plane lamina Applications of integration to he hydrostatic pressure on a plate Moments of inertia Sequences Infinite numerical series Power series Taylor and Maclaurin series Taylor's theorem for functions of two variable: stationary points and their identification Fourier series Determinants Matrices Matrix multiplication The inverse matrix Solution of a system of linear equations: Gaussian elimination The Gauss-Seidel iterative method The algebraic eigenvalue problem Scalars, vectors and vector addition Vectors in component form The straight line The scalar product (dot product) The plane The vector product (cross product) Applications of the vector product Differentiation and integration of vectors Dynamics of a particle and the motion of a particle in a plane Scalar and vector fields and the gradient of a scalar function Ordinary differential equations: order and degree, initial and boundary conditions First order differential equations solvable by separation of variables The method of isoclines and Euler's methods Homogeneous and near homogeneous equations Exact differential equations The first order linear differential equation The Bernoulli equation The structure of solutions of linear differential equations of any order Determining the complementary function for constant coefficient equations Determining particular integrals of constant coefficient equations Differential equations describing oscillations Simultaneous first order linear constant coefficient different equations The Laplace transform and transform pairs The Laplace transform of derivatives The shift theorems and the Heaviside step function Solution of initial value problems The Delta function and its use in initial value problems with the Laplace transform Enlarging the list of Laplace transform pairs Symbolic Algebraic Manipulation by Computer Software Answers References

show more