Module Information
Course Delivery
Assessment
| Assessment Type | Assessment length / details | Proportion |
|---|---|---|
| Semester Assessment | 1.5 Hours Test 1 Assessment of basic Algebra Skills and Differential Calculus skills. | 15% |
| Semester Assessment | 1.5 Hours Test 2 Assessment of Integral Calculus skills and ability to solve first-order differential equations | 15% |
| Semester Exam | 3 Hours Semester Exam Final examination (written). | 70% |
| Supplementary Assessment | 1.5 Hours Test 2 | 15% |
| Supplementary Assessment | 1.5 Hours Test 1 | 15% |
| Supplementary Exam | 3 Hours Semester Exam | 70% |
Learning Outcomes
On successful completion of this module students should be able to:
Apply differential calculus to solve simple problems in physics and engineering
Apply integral calculus to solve simple problems in physics and engineering
Show familiarity with widely used Taylor series expansion occurring in Physics and Engineering
Demonstrate how to correct apply complex numbers in Physics and Engineering
Perform basic matrix manipulations and demonstrate their application in Physics and Engineering
Brief description
This module covers the basic algebra and calculus needed to study physical concepts and processes quantitatively. It introduces the basic mathematical skills needed to pursue studies in Physics and Engineering with examples of applications..
Aims
To equip students with key concepts from algebra (complex numbers, polynomials and functions) and from calculus (both differential and integral) which are needed to understand and model physical and engineering situations. To familiarize Physics and Engineering students with frequently occurring mathematical concepts and expressions that will underpin their studies. To enable students to formulate strategies for tackling solve physical problems and to solve them for key problems.
Content
Geometric Basics, Expanding brackets
Powers and Logarithms
Planar geometry: scalars and vectors.
2. Polynomials
Identify features: degree and leading coefficients, roots, symmetric/anti-symmetry
The relations between roots of a polynomial and its coefficients
Rational polynomials, Partial fractions, Horner’s method for Polynomial division.
3. Applied Differential Calculus
Rates of change of physical variables, velocity, acceleration
Geometric and physical applications of methods of differentiation
4. Elementary functions in physics and engineering
Polynomial, exponential, logarithmic, trigonometric, hyperbolic etc., periodic functions, even and odd functions.
Trigonometric and Hyperbolic identities
5. Applied Integral Calculus
Integral calculus developed and applied to physical situations: work, energy, power
Mass, density distributions, centre of mass; P Physical applications
Pressure and hydrostatics; Motion from acceleration functions; (all explicitly listed as physical applications in calculus outcomes)
6. Complex numbers
Algebraic properties;
Geometric representation in the Complex Plane;
Euler’s Formula,
Roots of Unity,
Applications to wave theory
7. Series
Taylor series expansion with application to physical modelling
Convergence issues
8. Ordinary Differential Equations
Classifying differential equations: Order, ordinary vs. partial, homogeneity, linearity
First-order equations with separable variables. Radioactive decay, Newton’s law of Cooling, Falling under air resistance, Boundary conditions (e.g. initial values).
Homogeneous linear first-order equations. Integrating factor method.
Equilibrium solutions and their stability.
Non-homogeneous equations. Particular function. Driven oscillations.
Linear ODE with constant coefficients. Characteristic polynomial. Under and over damping
Physical applications: Standing waves, mechanical and electrical oscillators.
9. Introduction to Matrix Theory
Linear transformations in the plane
2×2 Matrices: identity, composition, determinant, inversion, solving simultaneous equations
Eigenvectors and eigenvalues
Primer for n×n matrices
Applications to Physics and Engineering
Module Skills
| Skills Type | Skills details |
|---|---|
| Application of Number | Application of numbers occurs in examples. |
| Communication | Students will have to state definitions of mathematical terms concisely. |
| Improving own Learning and Performance | There is opportunity to learn from feedback in the workshops and so to improve perfromance. |
| Problem solving | Mathematical problems to be solved in each of the workshops. |
| Research skills | Research skills are developed through background reading on the module topics |
| Subject Specific Skills | Translating physical problems into mathematical equations and models. |
| Team work | There is opportunity for group work in the workshops where students are encouraged to work together to solve problems and learn from each other. |
Notes
This module is at CQFW Level 4