Metric Homepage

Algebra

Algebra. To factorise quadratics and solve quadratic equations, to simplify rational expressions and resolve into partial fractions, to calculate terms and sums of arithmetic and geometric series.

Complex Numbers

To add, subtract, multiply and divide complex numbers in the form x + i y. To solve polynomial equations with complex roots. To calculate the conjugate, modulus and argument of complex numbers. To use de Moivre's Theorem to calculate trigonometric identities, and the nth roots of complex numbers. To calculute exponential and logarithmic functions of complex numbers.

Coordinate Geometry

To relate the equation of a straight line to its gradient and intercept, to relate the equation of a circle to its centre and radius.

Differential Equations

To find general and particular solutions of first order differential equations that have separable variables, or are exact or linear. To find general and particular solutions of second-order linear differential equations with constant coefficients. To find solutions of more general second-order differential equations in series form. To locate and classify critical points of coupled systems.

Differentiation

To calculate, from first principles, derivatives and gradients at a point, to calculate derivatives of standard functions and their sums, products, quotients and composites, to use differentiation to locate and classify stationary points, and to locate points of inflexion, to calculate series expansions by repeated differentiation.

Fourier Theory

To calculate Fourier series, and sine and cosine half-range series, of simple periodic functions.

Functions and Graphs

To express piecewise functions in terms of the Heaviside step function, to solve inequalities based on the modulus function, to compose and invert functions, to recognise odd and even functions, to perform calculations using exponential, logarithmic and hyperbolic functions.

Integration

To calculate integrals using the techniques of substitution and integration by parts, to integrate rational functions by resolving into partial fractions.

Laplace Transforms

To calculate the Laplace transforms of powers of t, as well as exponential and trigonometric functions. To use the Shift Theorems, together with known results, to calculate inverse Laplace transforms. To solve linear second-order differential equations, with constant coefficients, using the method of Laplace transforms.

Limits

To calculate limits of functions as x tends to some finite value, or to plus or minus infinity, if necessary with the aid of l'Hopital's Rule.

Matrices

To calculate determininants and use them to solve equation systems; to invert square matrices using cofactors; to solve equation systems and invert square matrices by Gaussian elimination; to calculate eigenvalues and eigenvectors and thus diagonalise square matrices; to invert square matrices using the Cayley-Hamilton theorem.

Mechanics

To calculate speeds, distances and times under uniform acceleration; to analyse the motion of projectiles; to calculate motion relative to a moving frame of reference; to analyse the forces on a free particle or body; to resolve a force into its components; to calculate resultant forces; to predict the motion of systems of pulleys; to calculate the centre of mass of a compound body; to balance forces and moments on a rigid body in equilibrium.

Numbers and Arithmetic

To carry out prime factorisations, manipulate indices and surds and rearrange inequalities.

Numerical Methods

To use numerical algorithms to calculate integrals, including the use of the Richardson extrapolation. To solve algebraic equations by iteration, including Newton's method. To fit curves to data by regression or interpolation. To solve simple differential equations approximately using Euler's method.

Partial Differentiation

To calculate partial derivatives of simple functions. To use partial differentiation to calculate relative and absolute errors. To calculate higher partial derivatives. To use partial differentiation to locate and classify stationary points of functions of two variables.

Permutations and Combinations

Combinatorics. To calculate permutations and combinations, to use the Binomial Theorem, for integer exponents, to calculate expansions.

Statistics

To calculate the probabilities of one-off events and combinations of events; to recognise and use mutual exclusion and independence; to calculate means and medians, and estimate them for grouped continuous data; to calculate standard deviations and interquartile ranges, and estimare them for grouped continuous data; to calculate probabilities based on the binomial and Poisson distributions; to use tables to perform calculations based on the Normal distribution; to set up null and alternative hypotheses in a range of scenarios; to test hypotheses at a range of significance levels.

Trigonometry

To express important angles in radians, and calculate their sines, cosines and tangents, to calculate the domain, range and period of trigonometric functions and their inverses, to calculate areas of plane figures, to use the pythagorea, addition, subtraction and double-angle identities.

Vectors

To calculate the magnitude of a vector, and the scalar product of two. To calculate the unit vector parallel to a given vector. To calculate the vector equation of a line. To calculate the angle between two vectors or lines. To calculate the vector equation of a plane. To calculate distances among points, lines and planes. To calculate the scalar triple product of three vectors. To use mathematic...more

Explorations

Explore the underlying concepts in trigonometry, coordinate geometry, differentiation, integration, partial differentiation and numerical methods.

Toolbox

Plotting tools.


METRIC 'Taster' METRIC comprises a bank of online mathematical resources covering a variety of topic areas for GCSE, 1st and 2nd year undergraduate courses in science and engineering. The application, developed by Imperial College London, was originally designed as a stand-alone, internet resource; however, in order to increase its functionality and overall usefulness, the current version has been fully integrated into Imperial's Virtual Learning Environment. The METRIC application can be divided into 4 areas; Theory Visualisation Demonstration and exploration of concepts Interactive self-test exercises The following sections highlight the current topic areas within Metric with limited functionality within each topic area, so that you can get a feel for the type of content in Metric and how it is treated within the application. If you require any further information, please do not hesitate to contact the metric team : metric@imperial.ac.uk
Algebra Algebra. To factorise quadratics and solve quadratic equations, to simplify rational expressions and resolve into partial fractions, to calculate terms and sums of arithmetic and geometric series.	Complex Numbers To add, subtract, multiply and divide complex numbers in the form x + i y. To solve polynomial equations with complex roots. To calculate the conjugate, modulus and argument of complex numbers. To use de Moivre's Theorem to calculate trigonometric identities, and the nth roots of complex numbers. To calculute exponential and logarithmic functions of complex numbers.
Coordinate Geometry To relate the equation of a straight line to its gradient and intercept, to relate the equation of a circle to its centre and radius.	Differential Equations To find general and particular solutions of first order differential equations that have separable variables, or are exact or linear. To find general and particular solutions of second-order linear differential equations with constant coefficients. To find solutions of more general second-order differential equations in series form. To locate and classify critical points of coupled systems.
Differentiation To calculate, from first principles, derivatives and gradients at a point, to calculate derivatives of standard functions and their sums, products, quotients and composites, to use differentiation to locate and classify stationary points, and to locate points of inflexion, to calculate series expansions by repeated differentiation.	Fourier Theory To calculate Fourier series, and sine and cosine half-range series, of simple periodic functions.
Functions and Graphs To express piecewise functions in terms of the Heaviside step function, to solve inequalities based on the modulus function, to compose and invert functions, to recognise odd and even functions, to perform calculations using exponential, logarithmic and hyperbolic functions.	Integration To calculate integrals using the techniques of substitution and integration by parts, to integrate rational functions by resolving into partial fractions.
Laplace Transforms To calculate the Laplace transforms of powers of t, as well as exponential and trigonometric functions. To use the Shift Theorems, together with known results, to calculate inverse Laplace transforms. To solve linear second-order differential equations, with constant coefficients, using the method of Laplace transforms.	Limits To calculate limits of functions as x tends to some finite value, or to plus or minus infinity, if necessary with the aid of l'Hopital's Rule.
Matrices To calculate determininants and use them to solve equation systems; to invert square matrices using cofactors; to solve equation systems and invert square matrices by Gaussian elimination; to calculate eigenvalues and eigenvectors and thus diagonalise square matrices; to invert square matrices using the Cayley-Hamilton theorem.	Mechanics To calculate speeds, distances and times under uniform acceleration; to analyse the motion of projectiles; to calculate motion relative to a moving frame of reference; to analyse the forces on a free particle or body; to resolve a force into its components; to calculate resultant forces; to predict the motion of systems of pulleys; to calculate the centre of mass of a compound body; to balance forces and moments on a rigid body in equilibrium.
Numbers and Arithmetic To carry out prime factorisations, manipulate indices and surds and rearrange inequalities.	Numerical Methods To use numerical algorithms to calculate integrals, including the use of the Richardson extrapolation. To solve algebraic equations by iteration, including Newton's method. To fit curves to data by regression or interpolation. To solve simple differential equations approximately using Euler's method.
Partial Differentiation To calculate partial derivatives of simple functions. To use partial differentiation to calculate relative and absolute errors. To calculate higher partial derivatives. To use partial differentiation to locate and classify stationary points of functions of two variables.	Permutations and Combinations Combinatorics. To calculate permutations and combinations, to use the Binomial Theorem, for integer exponents, to calculate expansions.
Statistics To calculate the probabilities of one-off events and combinations of events; to recognise and use mutual exclusion and independence; to calculate means and medians, and estimate them for grouped continuous data; to calculate standard deviations and interquartile ranges, and estimare them for grouped continuous data; to calculate probabilities based on the binomial and Poisson distributions; to use tables to perform calculations based on the Normal distribution; to set up null and alternative hypotheses in a range of scenarios; to test hypotheses at a range of significance levels.	Trigonometry To express important angles in radians, and calculate their sines, cosines and tangents, to calculate the domain, range and period of trigonometric functions and their inverses, to calculate areas of plane figures, to use the pythagorea, addition, subtraction and double-angle identities.
Vectors To calculate the magnitude of a vector, and the scalar product of two. To calculate the unit vector parallel to a given vector. To calculate the vector equation of a line. To calculate the angle between two vectors or lines. To calculate the vector equation of a plane. To calculate distances among points, lines and planes. To calculate the scalar triple product of three vectors. To use mathematic...more
Explorations Explore the underlying concepts in trigonometry, coordinate geometry, differentiation, integration, partial differentiation and numerical methods.	Toolbox Plotting tools.