Department of Physics, Unit Catalogue 2008/09 |
PH10007 Mathematics for scientists 1 |
| Credits: 6 |
| Level: Certificate |
| Semester: 1 |
| Assessment: EX 100% |
| Requisites: |
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Students must have A-level Mathematics to undertake this unit.
Aims: The aim of this unit is to introduce basic mathematical techniques required by science students, both by providing a reinterpretation of material already covered at A-level in a more general and algebraic form and by introducing more advanced topics.
Learning Outcomes: After taking this unit the student should be able to: * sketch graphs of standard functions and their inverses; * evaluate the derivative of a function and the partial derivative of a function of two or more variables; * write down the Taylor series approximation to a function; * represent complex numbers in Cartesian, polar and exponential forms, and convert between these forms; * calculate the magnitude of a vector, and the scalar and vector products of two vectors; * solve simple geometrical problems using vectors. Skills: Numeracy T/F A, Problem Solving T/F A. Content: Functions of a real variable (3 hours): Graphs of standard functions (polynomial, exponential, logarithmic, trigonometric and hyperbolic). Domains and ranges. Composite functions. Inverse functions. Symmetries and transformations (reflections, rotation) of graphs. Differentiation (9 hours): Limits and continuity, differentiability. Review of differentiation. Higher derivatives, meaning of derivatives. Graphical interpretation of derivatives. Logarithmic, parametric and implicit derivatives. Taylor and Maclaurin expansions, standard series. Convergence of series; ratio test, limits, L'Hopital's rule. Numerical differentiation. Functions of two variables. Partial differentiation. Taylor expansion in two variables. Chain rule. Small changes and differentials, total derivative. Complex numbers (4 hours): Definition and algebra of complex numbers in x+iy form. Complex conjugate. Modulus and argument. Argand diagram, r exp(iθ) form. De Moivre's theorem. Solution of equations involving complex variables. Vector algebra (6 hours): Introduction to vectors; physical examples of scalar and vector quantities. Magnitude of a vector, unit vector. Cartesian components. Scalar product; projections, components, physical examples. Vector product; determinantal form for Cartesian components, physical examples. Vector definitions of lines and planes. Triple product. Introduction to vector spaces. |