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See why the field inside a conductor is zero and where its charge sits, understand dielectrics and polarisation, derive the capacitance of a parallel plate capacitor with and without a dielectric, and combine capacitors in series and parallel and find the energy stored.
Read articleDefine electric potential and potential difference and find the potential of a point charge, add potentials for a dipole and a system of charges, relate equipotential surfaces to the field, and calculate the potential energy of charges and of a dipole in a field.
Read articleCalculate the electric flux of a uniform field through a surface, find the field of an electric dipole on its axial and equatorial lines, derive the torque on a dipole in a uniform field, and use Gauss's law for a line charge, a plane sheet and a spherical shell.
Read articleLearn the basic properties of charge — quantisation, conservation and additivity — and how conductors differ from insulators, apply Coulomb's law in vector form, add forces by superposition, and calculate electric fields and read field line patterns.
Read articleIdentify a partition of the sample space and state the theorem of total probability, use it across mutually exclusive and exhaustive causes, apply Bayes' theorem to update prior probabilities, and interpret posterior probabilities in real-life problems.
Read articleCompute the conditional probability P(E|F) from a sample space and use its properties, apply the multiplication theorem to two or more events, and test events for independence while keeping independence separate from mutual exclusiveness.
Read articleLearn the language of linear programming — constraints, objective function and feasible solutions — turn real problems into linear programming problems, sketch and classify feasible regions, and find optimal solutions at corner points, including when an unbounded region has none.
Read articleFind the angle between two lines in vector or Cartesian form and the conditions for perpendicular and parallel lines, recognise skew lines, compute the shortest distance between skew lines, and find the distance between parallel lines.
Read articleDefine direction cosines and direction ratios and prove l^2 + m^2 + n^2 = 1, find them for the line through two points, and write vector and Cartesian equations of a line through a point parallel to a vector or through two points.
Read articleCompute the dot product and read its geometry, use it for angles, perpendicularity and projections, compute the cross product and see why it is not commutative, and apply it to areas of triangles and parallelograms and to test parallel vectors.
Read articleTell scalars from vectors and find magnitude, direction cosines and direction ratios, recognise equal, unit, zero, collinear and negative vectors and write position vectors, add vectors by the triangle and parallelogram laws, and apply the section formula.
Read articleRecognise homogeneous first order equations and solve them with y = vx, solve linear equations dy/dx + Py = Q using an integrating factor, handle the dx/dy + P1x = Q1 form, and find particular solutions from given conditions.
Read articleFind the order and degree of a differential equation and see when the degree is not defined, tell general solutions from particular ones and verify solutions, and solve first order equations by separating the variables and using initial conditions.
Read articleSet up definite integrals for the area between a curve and an axis, find areas enclosed by lines, circles, parabolas and ellipses in standard form, and handle regions that dip below the axis or must be split into parts.
Read articleDefine the definite integral with its limits and state the Fundamental Theorem of Calculus, evaluate definite integrals with antiderivatives, change limits correctly when substituting, and use additive, reversal, even-odd and complementary-limit properties to simplify.
Read articleUse the special integrals of forms like 1/(x^2 + a^2) and their square-root versions, split rational functions into partial fractions, apply integration by parts including the e^x[f(x) + f'(x)] pattern, and integrate square roots of quadratic expressions.
Read articleSee integration as the reverse of differentiation and why every indefinite integral carries a constant, apply standard formulae and properties, evaluate integrals by substitution, and simplify trigonometric integrands with identities before integrating.
Read articleTell local maxima and minima from absolute extreme values and find critical points, classify them with the first and second derivative tests, and solve optimisation problems, including absolute extrema on a closed interval.
Read articleRead a derivative as a rate of change and solve related rates problems, use the sign of the first derivative to find where a function strictly increases or decreases, and apply that behaviour to changing areas, volumes and costs.
Read articleUnderstand exponential and logarithmic functions and differentiate e^x and log x, use logarithmic differentiation for powers like x^x and long products, find dy/dx for curves in parametric form, and compute second order derivatives to verify relations.
Read articleTest continuity at a point using left-hand and right-hand limits and locate discontinuities, apply the algebra of continuous functions, use the chain rule and see why differentiability implies continuity, and differentiate inverse trigonometric and implicit functions.
Read articleBuild the adjoint of a square matrix and verify A(adj A) = |A|I, tell singular from non-singular matrices and compute inverses, decide whether a linear system is consistent, and solve systems with a unique solution by the matrix method.
Read articleEvaluate 2 by 2 and 3 by 3 determinants by expanding along any row or column, find minors and cofactors and use them to expand, and apply determinants to the area of a triangle and to test whether three points are collinear.
Read articleMultiply conformable matrices and use their associative and distributive laws, see why AB and BA differ and how two non-zero matrices can multiply to zero, apply transpose rules, and split any square matrix into symmetric and skew symmetric parts.
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