PH12003: Mathematics for physics 1
[Page last updated: 15 August 2024]
| Academic Year: | 2024/25 |
| Owning Department/School: | Department of Physics |
| Credits: | 15 [equivalent to 30 CATS credits] |
| Notional Study Hours: | 300 |
| Level: | Certificate (FHEQ level 4) |
| Period: |
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| Assessment Summary: | EXCB 50%, EXOB 50% |
| Assessment Detail: |
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| Supplementary Assessment: |
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| Requisites: |
While taking this module you must take PH12002
You must have A-level Mathematics (or equivalent) to take this module. |
| Learning Outcomes: |
After taking this unit the student should be able to:
evaluate the derivative of a function and the partial derivative of a function of two or more variables;
analyse stationary points and apply this for problem solving;
integrate functions using a variety of standard techniques;
apply discrete and continuous probability distributions to find probabilities of events, expected values and variances;
write down or derive 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;
use multiple integrals to find areas, volumes, and simple physical properties of solids;
find the general solution of first and second order ordinary differential equations and show how a particular solution may be found using boundary conditions;
calculate the determinant and inverse of a matrix, and the product of two matrices;
use matrix methods to solve simple linear systems;
construct simple mathematical models to solve problems in physics;
appraise the suitability of known methods for mathematical modelling;
demonstrate the utility of transform methods for solving differential equations;
outline the advantages and disadvantages of numerical solutions of ordinary differential equations |
| Synopsis: | Physics and mathematics are disciplines with a natural affinity; mathematics lies at the heart of our understanding of the physical world, and the study of the equations of physics has motivated large areas of mathematical analysis. You will learn the core mathematical techniques required to study physics and explore how these can be applied to physical problems. |
| Content: | Vector algebra
Multiplication of vectors [vector product; scalar triple product; vector triple product], Equations of lines, planes and sphere. Using vectors to find distances (Point to line; point to plane; line to line; line to plane; plane polar coordinates). Reciprocal vectors. Vector Area.
Complex numbers and hyperbolic functions
Manipulation of complex numbers [Addition and subtraction; modulus and argument; multiplication; complex conjugate; division], Polar representation of complex numbers [Multiplication and division in polar form], de Moivre's theorem [Trigonometric identities; finding the nth roots of unity; solving polynomial equations], Applications to differentiation and integration, Hyperbolic functions [Definitions; hyperbolic-trigonometric analogies; identities of hyperbolic functions; solving hyperbolic equations; inverses of hyperbolic functions; calculus of hyperbolic functions].
Series and limits
Series, Operations with series, Power series [Convergence of power series; operations with power series], Taylor series [Taylor's theorem; approximation errors; standard Maclaurin series], Evaluation of limits [L'Hopital's rule]
Partial differentiation
Definition of the partial derivative, The total differential and total derivative, Exact and inexact differentials, The chain rule, Taylor's theorem for many-variable functions, stationary values of many-variable functions.
Integration
Integration from first principles; the inverse of differentiation; infinite and improper integrals; parts, partial fractions, completing the square, tan-half-angle
Multiple integrals
Double integrals, Triple integrals, Applications of multiple integrals [Areas and volumes; masses, centres of mass and centroids; mean values of functions], Change of variables in multiple integrals.
Probability and distributions
Discrete distributions [Poisson distribution, mean and variance; expectation values], Continuous distributions [expectation values, Gaussian distribution including as an approximation for Binomial and Poisson; simple applications, e.g. velocity distributions], Central limit theorem.
Ordinary differential equations
_[Basics]_ General form of solution, First-degree first-order equations [Separable-variable equations; exact equations; inexact equations, integrating factor for inexact equations.]. Higher-order ordinary differential equations. Linear equations with constant coefficients [Finding the complementary function yc(x); finding the particular integral yp(x); constructing the general solution yc(x) + yp(x).
_[Further]_ Linear first degree first order ODEs with integrating factor. Bernoulli equations. The logistic equation.
Modelling
Mathematisation of systems described by text. Test elements and applying physical laws [Conservation of number, Newton's third law, Rates of change].
Linear Algebra
_[Basics]_ Linear operators, Matrices, Basic matrix algebra [Matrix addition; multiplication by a scalar; matrix multiplication], The transpose of a matrix, The trace of a matrix, The determinant of a matrix [Properties of determinants], The inverse of a matrix. Eigenvectors and eigenvalues and their determination [Degenerate eigenvalues].
_[Further]_ Special types of square matrix [Diagonal; triangular; symmetric and antisymmetric; orthogonal; normal]. Applications of eigenvectors. Typical oscillatory systems, Change of basis, Diagonalization of matrices, Quadratic forms, Simultaneous linear equations.
Introduction to Analysis
Sets: definitions; notation; subsets and intervals; cartesian product; completeness of the reals. Functions: definition; injections, surjections and bijections; cardinality and countability of sets. Proof: examples of direct proof, proof by contradiction and proof by induction. Sequences: definitions; ε-N approach to convergence and limits; Cauchy convergence criterion; convergence of infinite series. Function limits, continuity and differentiability: limit points; ε-δ metthod; limits and continuity; differentiation and differentiability; Rolle's theorem, mean-value theorem and Taylor's theorem.
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| Course availability: |
PH12003 is a Must Pass Unit on the following courses:Department of Physics
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Notes:
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