Calculus and Vectors A
|Unit level:||Level 1|
|Teaching period(s):||Semester 1|
|Offered by||School of Mathematics|
|Available as a free choice unit?:||N
The course unit unit aims to provide a firm foundation in the concepts and techniques of the calculus, including real and complex numbers, standard functions, curve sketching, Taylor series, limits, continuity, differentiation, integration, vectors in two and three dimensions and the calculus of functions of more than one variable.
The unit introduces the basic ideas of complex numbers relating them to the standard rational and transcendental functions of calculus. The core concepts of limits, differentiation and integration are revised. Techniques for applying the calculus are developed and strongly reinforced. Vectors in two and three dimensions are introduced and this leads on to the calculus of functions of more than one variable, vector calculus, integration in the plane, Green's theorem, Stokes' theorem and Gauss' theorem.
On successful completion of this module students will be able to:
- Manipulate complex numbers and use them to relate trigonometric, hyperbolic and exponential functions.
- Evaluate complex roots and represent complex numbers in the complex plane, using results such as Euler's formula and de Moivre's theorem.
- Sketch polynomial, rational, inverse and some standard functions of a single variable, expressed in Cartesian or polar coordinates.
- Evaluate and interpret limits and derivatives of algebraic functions, including functions expressed in implicit or parametric form.
- Select and deploy methods for evaluating integrals of functions of a single variable.
- Construct and manipulate Taylor series of scalar functions of one and two variables.
- Manipulate vectors using tools such as scalar and vector products, deploying these quantities to solve geometric problems.
- Construct, deploy and interpret derivatives of scalar- and vector-valued functions of more than one variable.
- Construct, evaluate and interpret simple integrals of functions of two variables, using tools such as the Jacobian and Green's theorem.
- Construct and evaluate line integrals for conservative and non-conservative vector fields.
- Locate and classify extrema of functions of two variables, using discriminants and Lagrange multipliers.
- Other - 20%
- Written exam - 80%
Assessment Further Information
Supervision attendance and participation; Weighting within unit 10%
Coursework; In-class test in week 6, weighting within unit 10%
Three hours end of semester examination; Weighting within unit 80%
- Foundations: real numbers, inequalities, "infinity", intervals, modulus, square root; imaginary numbers; complex numbers, standard form, complex plane, polar form, modulus, complex conjugate, algebra of complex numbers; functions, graphs, sketches of simple functions, basic idea of limits, domain and range, combining functions, simple transformations, inverse functions; trigonometric, exponential, logarithmic and hyperbolic functions and their inverses; symmetry, periodicity, increasing, decreasing, monotonicity.
- Limits and differentiation: notions of limits and continuity; discontinuities, left and right limits; limits of sums, products, quotients, compositions, l'Hôpital's rule; order notation; definition of derivative and higher derivatives; derivatives of inverse functions; derivatives of trigonometric functions, exponential, logarithm, powers, hyperbolic functions and their inverse functions; sums, products, quotients and chain rule; derivatives of implicit and parametric functions; logarithmic differentiation; turning and critical points; curves at right angles.
- Power series: notion of a power series as a limit of polynomials, its derivative and integral; Taylor series; radius of convergence (not proven); series for exponential, sine, cosine and other functions; manipulating Taylor expansions.
- More on complex numbers: Euler's formula; complex forms of sine and cosine, relationship between trigonometric and hyperbolic identities; de Moivre's theorem; exponential and polar forms of complex numbers; complex roots.
- Integration: definite and indefinite integrals; fundamental theorem of calculus; proper and improper integrals; singularities in definite integrals; techniques for integration: standard integrals, function of a linear function, integral of a sum, algebraic manipulation, integration by parts, recurrence formulae, substitutions and their identification, partial fractions; area between curves (including polar coordinates), lengths of curves (including polar coordinates), surfaces and volumes of revolution.
- Curve sketching: Cartesian and polar coordinates; parametric curves; generic conic sections; powers (with positive and negative integer exponent); factorised polynomials and rational functions, zeros, singularities, asymptotic behaviour, turning or critical points; simple curves in polar coordinates.
- Vectors in two and three dimensions: points and vectors; vectors as directed line segments (magnitude, direction); Cartesian representation, coordinates; addition, subtraction, multiplication by a scalar, magnitude, unit vectors, inner/dot product, angle between two vectors, orthogonality, components or projections, extension to higher dimensions; vector/cross product in three dimensions, its magnitude and direction (orthogonality), cross product as a "turning effect"; scalar triple product; vector representation of points, lines and planes, and relations between them; orthogonal coordinate systems (polar, parabolic, elliptic, cylindrical and spherical).
- Functions of more than one variable: partial derivative (definition, notation and evaluation), chain-rule, higher partial derivatives including mixed derivatives, higher dimensional Taylor expansion; gradient vector (grad); critical or turning points (maxima, minima, saddle-points), relationship with contour lines and grad, identifying critical points, derivative test and use of the discriminant; turning points with constraints, Lagrange multipliers; div, curl and related identities.
- Multiple integrals: area integration of a scalar-valued function in the plane, choice of order of integration; area integral in polar coordinates; Jacobian and change of variable; directed path or line integral of a vector valued function in the plane, path-dependence/independence, conservative vector fields; potential function
James Stewart, Calculus, Early Transcendentals, International Student Edition, Thomson (any recent edition).
[This text covers almost every aspect of what you will be learning, with many examples. You should ensure that you can have easy access to a copy.
Useful background material can be found in:
Hugh Neill and Douglas Quadling. Cambridge Advanced Mathematics Core 3 and 4.
[This text describes well and clearly what should be known from A-level. It (as for other A-level texts) provides an introduction to what should be known before entering a university calculus course.]
Feedback seminars will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
- Lectures - 44 hours
- Tutorials - 11 hours
- Independent study hours - 145 hours