The module develops the learner’s abilities in advanced calculus by emphasising the methodologies of solving problems in multivariate calculus and integration.
Differentiation: functions of one and several variables; interpretation of the derivative and partial derivatives – extrema and saddle points; higher order derivatives and their interpretation; rules of differentiation; L’Hopital’s rule; Chain rule in one and several variables. Taylor’s theorem and Taylor series in one and several variables. Examples taken from mechanics, polar coordinates etc.;
Integration: statement of fundamental theorem of calculus; techniques of integration (substitution: half-angle substitutions, reduction formulae); double integrals and integration over the plane; change of variables and the Jacobian; examples and physical interpretation.
Ordinary Differential Equations: classification of differential equations (order, linearity, homogeneous); first-order equations: homogeneous, separable, integrating factor, exact; second-order equations: linear, constant coefficients. Examples and applications: e.g. population growth models, compartmental models, electrical circuits, simple and forced harmonic motion and other physical problems.
Cartesian Vector Calculus: vector valued functions; gradient; directional derivatives; divergence and curl of a vector field; examples and identities.
Lectures supported by tutorials and/or laboratory sessions including use of mathematical software.
| Module Content & Assessment | |
|---|---|
| Assessment Breakdown | % |
| Formal Examination | 70 |
| Other Assessment(s) | 30 |