Taylor series in 2D up to quadratic terms, classification of critical points. Application of chain rule to coordinate transformations. Implicit differentiation: of functions of one variable and of scalar fields tangent lines to level curves. The gradient vector: geometric interpretation, directional derivative, tangent planes. Equality of mixed second partial derivatives, chain rule. Tangent plane as linear approximation to the surface at a point. Directional derivatives derived from chain rule. Partial derivatives: intuitive notion, statement of chain rule, examples. "Examples, including the evaluation of series.įunctions of two variables. Periodic extension, convergence and a statement of Dirichlet's theorem. Fourier coefficients, even and odd functions. Trigonometric and complex exponential forms. Standard examples: trigonometric and hyperbolic functions, exponential and logarithmic series, binomial series and relation with binomial theorem. Taylor series with integral form of remainder, statement of derivative form. Second-order: reduction of order linear with constant coefficients, homogeneous and inhomogeneous. First-order: separable, linear (by integrating factor). Classification, existence and uniqueness of solutions (Lipschitz-Picard condition), constants of integration. Relationships between trigonometric and hyperbolic functions, connections with Algebra: the complex numbers, Euler’s formula. Conic sections as polynomial equations of degree 2 in two variables. Rules and techniques for integration: partial fractions, by parts, by substitution. Anti-derivatives and the indefinite integral. Derivatives of inverse functions via chain rule (and thus dy/dx=1/(dx/dy)). and exp(x)=e x as the limit of Euler’s sequence.The exponential function: as solution of dy/dx=y with y(0)=1 as a power series, leading to e=1+1/2+1/3! + Derivatives of standard functions (powers, polynomials, trigonometric). Rules of differentiation (chain, product, quotient). Limits and continuity (axiomatic as definitions). In addition to the above broad aims of the first year, this module focusses on ensuring that students have competence in a wide range of essential concepts, techniques and applications of differential and integral calculus, and differential equations. This development is addressed in all of our first year modules, although different modules have a different emphasis. Students will develop both their knowledge of mathematics as a subject and their reasoning and communication skills, through lectures, tutorials, seminars, guided self-study, independent learning and project work. During first year, as well as consolidating, broadening and extending core material from pre-University study, we initiate a cultural transition to the rigorous development of mathematics which is characteristic at University.
The first years of all mathematics programmes are designed to give students a thorough grounding in a wide spectrum of mathematical ideas, techniques and tools in order to equip them for the later stages of their course. See module specification for other years:Īutumn Term 2020-21 to Summer Term 2020-21.
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