PH10007: Mathematical methods for physics 1
[Page last updated: 21 April 2022]
| Academic Year: | 2022/3 |
| Owning Department/School: | Department of Physics |
| Credits: | 12 [equivalent to 24 CATS credits] |
| Notional Study Hours: | 240 |
| Level: | Certificate (FHEQ level 4) |
| Period: |
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| Assessment Summary: | CW 30%, EX 70% |
| Assessment Detail: |
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| Supplementary Assessment: |
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| Requisites: | You must have A-level Physics (or equivalent) and A-level Mathematics (or equivalent) to take this unit |
| 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 probabilty 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 of two vectors; * solve simple geometrical problems using vectors. * Use mutilpe integrals to find areas, volumes and simple 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; * solve some first and second order partial differential equations by separation of variables; * calculate the determinant and inverse of a matrix, and the product of two matrices; * use matrix methods to solve simple linear systems. |
| Aims: | The aim of this unit is to introduce mathematical techniques required by physical science students for first year physics and to enable more advanced study.
The unit also aims to show regularly the application of mathematics to physical problems as well as underpinning mathematical ideas. |
| Skills: | Numeracy T/F A, Problem Solving T/F A. |
| Content: | : Preliminary calculus (8 hours, including problems classes)
Differentiation [Limits - continuity, Differentiation from first principles; product, chain, implicit and inverse, logarithmic differentiation; Higher derivatives - Leibnitz' theorem; radius of curvature] Integration [Integration from first principles; the inverse of differentiation; infinite and improper integrals; parts, tan-half-angle, reduction formulae, integral inequalities, improper integrals;] Probability and distributions (6 hours, including problems classes) Definition of probability, Permutations and combinations Discrete distributions [mean and variance; expectation values; Binomial and Poisson distributions] 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 (5 hours, including problems classes) 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] Complex logarithms and complex powers 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 (4 hours, including problems classes) Multiplication of vectors [vector product; scalar triple product; vector triple product] Equations of lines, planes and spheres Using vectors to find distances [Point to line; point to plane; line to line; line to plane; plane polar coordinates] Reciprocal vectors Series and limits (4 hours, including problems classes) 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 (6 hours, including problems classes) 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 (7 hours, including problems classes) Double integrals, Triple integrals Applications of multiple integrals [Areas and volumes; masses, centres of mass and centroids; Pappus' theorems; moments of inertia; mean values of functions] Change of variables in multiple integrals [Change of variables in double integrals; change of variables in triple integrals] First-order ordinary differential equations (7 hours, including problems classes) General form of solution First-degree first-order equations [Separable-variable equations; exact equations; inexact equations, integrating factors; linear equations; homogeneous equations; Bernoulli's equation; miscellaneous equations] Higher-degree first-order equations [Equations soluble for p; for x; for y;] Higher-order ordinary differential equations (4 hours, including problems classes) 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 recurrence relations; Matrices and determinants (5 hours, including problems classes) Linear operators, Matrices, Basic matrix aalgebra[Matrix addition; multiplication by a scalar; matrix multiplication] Functions of matrices, The transpose of a matrix The complex and Hermitian conjugates of a matrix The trace of a matrix The determinant of a matrix [Properties of determinants] The inverse of a matrix The rank of a matrix Special types of square matrix [Diagonal; triangular; symmetric and antisymmetric; orthogonal; Hermitian and anti-Hermitian; unitary; normal] Linear Algebra (5 hours, including problems classes) Eigenvectors and eigenvalues [Of a normal matrix; of Hermitian and anti-Hermitian matrices; of a unitary matrix; of a general square matrix] Determination of eigenvalues and eigenvectors [Degenerate eigenvalues] Typical oscillatory systems Change of basis and similarity transformations Diagonalization of matrices Quadratic and Hermitian forms [Stationary properties of the eigenvectors; quadratic surfaces] Simultaneous linear equations [Range; null space; N simultaneous linear equations in N unknowns; singular value decomposition Modelling (5 hours, including problems classes) Mathematisation of systems described by text. Test elements and applying physical laws. e.g. conservation of number, Newton's third law, Rates of change. |
| Programme availability: |
PH10007 is a Designated Essential Unit on the following programmes:Department of Physics
PH10007 is Compulsory on the following programmes:Programmes in Natural Sciences
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