Biomedical Engineering Year 1
2008
Mathematics BME 102
Autumn Term
35 Lectures (Prof A Corti, 673 Huxley Building, a.corti@imperial.ac.uk)
Mondays at 4, Thursdays at 9 and 4 (plus Fridays at 11 and 2 in weeks 2-4).
SYLLABUS
ANALYSIS: Functions of one variable : exponential, logarithmic and trigonometric functions: odd, even,
inverse functions. Limits : continuous and discontinuous functions. Differentiation : implicit and logarithmic
differentiation ; Leibniz’s formula ; stationary points and points of inflection ; curve sketching ; polar
coordinates. Taylor’s and Maclaurin’s series ; l’Hopital’s rule . Convergence of power series ; ratio test ;
radius of convergence. Complex numbers : the complex plane ; polar representation ; de Moivre’s theorem
; ln z and exp z . Hyperbolic functions : inverse functions ; series expansions ; relations between
hyperbolic and trigonometric functions. Integration : definite and indefinite integrals ; the fundamental
theorem ; improper integrals ; integration by substitution and by parts ; partial fractions ; applications.
LINEAR ALGEBRA: Vector algebra : basic rules ; cartesian coordinates ; scalar and vector products ;
applications to geometry ; equations of lines and planes ; triple products ; linear dependence. Matrix algebra
: double suffix notation ; basic rules ; transpose, symmetric, diagonal, unit, triangular, inverse and
orthogonal matrices. Determinants : basic properties ; Cramer’s rule. Linear algebraic equations :
consistency ; elementary row operations ; linear dependence ; Gauss- Jordan method ; Gaussian
elimination ; LU factorisation. Eigenvalues and eigenvectors ; diagonalisation.
ORDINARY DIFFERENTIAL EQUATIONS: First order equations : separable, homogeneous, exact, linear.
Second order linear equations with constant coefficients.
FOURIER SERIES: Standard formulae ; even and odd functions ; half-range series ; complex form.
Parseval’s theorem. Differentiation and integration of series.
COURSE TOPICS ( Chronological order )
TOPIC
0
1
2
3
4
5
6
7
8
9
10
:
Review of Basic Material
Functions of one variable
Limits of functions
Differentiation
Taylor and Maclaurin series
Integration
Complex numbers ; hyperbolic functions
Ordinary differential equations
Vectors
6
Matrices and Linear Algebra
Fourier series
ASSESSMENT :
Approximate #
of Lectures
10
4
3
3
5
5
3
2
TERM
Aut
Aut
Aut
Aut
Aut
Aut
Aut
Aut
Spr
9
5
Spr
Spr
(i)
Examinations :
One 3-hour written exam, early in the Summer term,
consisting of 10 questions, with a rubric “Answer 8
questions“. Questions marked out of 15.
(ii)
Coursework
1 mastery test in Term 1, over the 10 Introductory
Lectures and 1 progress test.
:
These 2 exams in Term 1, together with 1 exam in Term 2
will contribute 10% towards the course assessment.
The mastery and progress tests in the Autumn term will take place on
November 14 (at 11) and December 12.
RECOMMENDED TEXTS :
(i) Glyn James
:
Modern Engineering Mathematics ; Addison-Wesley, 3rd edition, 2001
(ii) K A Stroud
Engineering Mathematics ; MacMillan , 5th edition , 2001
: