PH12003: Mathematics for physics 1
[Page last updated: 23 October 2023]
| Academic Year: | 2023/24 |
| 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: | Preliminary calculus
Differentiation [implicit and inverse, logarithmic differentiation; Higher derivatives; Leibnitz' theorem; radius of curvature]. Integration [infinite and improper integrals, tan-half-angle, reduction formulae, integral inequalities, improper 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.
Complex numbers and hyperbolic functions
The need for complex numbers, 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].
Vector algebra
Multiplication of vectors [vector product; scalar triple product; vector triple product], Equations of lines and planes, Using vectors to find distances.
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, Least squares fits.
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
Ordinary differential equations
General form of solution, First-degree first-order equations [Separable-variable equations; exact equations; inexact equations, integrating factors; linear 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).
Linear Algebra
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, Special types of square matrix [Diagonal; triangular; symmetric and antisymmetric; orthogonal; normal]. Eigenvectors and eigenvalues, Determination of eigenvalues and eigenvectors [Degenerate eigenvalues], Typical oscillatory systems, Change of basis, Diagonalization of matrices, Quadratic forms, Simultaneous linear equations.
Modelling
Mathematisation of systems described by text. Test elements and applying physical laws [Conservation of number, Newton's third law, Rates of change].
Laplace Transforms
Definition and notation, Transforms of simple functions, Properties of the Laplace transform, The inverse transform, solution of differential equations [ordinary differential equations, simultaneous differential equations.] Applications - electrical circuits and mechanical vibrations.
Numerical Solution of Ordinary Differential Equations
Euler's method, Multistep methods, local and global truncation errors, Runge-Kutta methods, numerical solutions of coupled first-order equations, state-space representation of coupled first-order equations. |
| Course availability: |
PH12003 is a Must Pass Unit on the following courses:Department of Physics
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Notes:
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