Algebra Functions Differentiation ODEs Discrete Mathematics Vectors Matrices 3D surfaces Complex Numbers Curves Trigonometry Number Systems Equations Inequalities Integration
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Algebra
Title and link  Description  Keywords  ID 
Multiplication of Letters and Numbers  Worked example explains how to multiply numbers and letters. Also explains the concepts of expressions, constants and coefficients.  Expressions, constants, coefficients  
Indices  Introduction to Indices. Definitions.Multiplication and Division of Indices  Indices, Multiplication, Division.  
Equation of a circle  Starting with an equation quadratic in x and y, this screencast shows the general form of a circle, to which the equation has to be converted. It then shows how to complete the square in x and in y and how to read off the centre and radius.  equation of circle, completing the square in x and y, centre, radius  S1 
Removing Brackets  Worked example shows how to remove brackets.  a(b+c), Multiplication  
Real Life Problems  Worked examples of where the algebra skills already learned can be used in real life  Real Life Problems, Formulas  
Adding Like Terms  Defines algebraic term, Gives examples of terms. Worked examples of simplifying expressions by adding like terms  Simplify, Expression, Terms, Variables  
Introduction to Algebra  Worked example shows how to solve an algebraic equation. Also explains the concepts of variable and operation.  Solving equations, operations and variable. 
Functions
Title and link  Description  Keywords  ID 
Orthogonal families of curves  Starting with the equation for a family of curves, this screencast shows how to differentiate both sides of the equation to produce a differential equation which can be solved to find the grandient of this family of curves. It is then shown how the general form of the gradient for the orthogonal family of curves can be obtained and hence the equation of the orthogonal curves.  orthogonal curves, first order differential equations, separable differential equations  S6 
Composition: What is it?  Explains the concept of composing functions. 
Differentiation
Title and link  Description  Keywords  ID 
first order partial differentiation  Two examples are given to demonstrate how to compute partial derivatives of a function of two variables.  partial derivatives, chain rule  S7 
Second order partial differentiation  This screencast defines the four secondorder partial derivatives that can be calculated for a function of two variables, and then demonstrates their calculation in practice for a specific example  partial derivatives, second order partial derivatives  S9 
Logarithmic differentiation (handwritten)  This screencast explains how differentiation requiring combinations (or multiple uses) of the product, chain and quotient rules can be made simpler by using logarithmic differentiation. This method relies on log rules to simplify a function before taking a derivative. This particular screencast shows handwritten step by step explanations. Please compare to the other screencast going through the same example but with typed explanations, and send us feedback on which explains this concept better.  logarithmic differentiation, log rules, product rule, chain rule, quotient rule, simplify differentiation  S23 
Logarithmic differentiation (typed)  This screencast explains how differentiation requiring combinations (or multiple uses) of the product, chain and quotient rules can be made simpler by using logarithmic differentiation. This method relies on log rules to simplify a function before taking a derivative. This particular screencast shows typed explanations. Please compare to the other screencast going through the same example but with handwritten explanations, and send us feedback on which explains this concept better.  logarithmic differentiation, log rules, product rule, chain rule, quotient rule, simplify differentiation  S47 
The Multivariate Chain Rule  Shows how to find a particular form of the chain rule that is needed, depending on the type of problem that is posed. Uses a tree diagram to develop the rule.  multivariate differentiation, chain rule, partial derivative  S48 
Multivariate Chain Rule 2  Example of using the multivariate chain rule where z=f(x,y) and x and y are both functions of t.  multivariate differentiation, chain rule, partial derivative  S54 
Chain Rule 2 variables  Applies the multivariate chain rule to an example where z = f(x,y), x = x(t), y = y(t), to find dz/dt.  multivariate chain rule, differentiation, two variables, partial derivative  S67 
Directional Derivatives  An example where the gradient of a surface is found depending on the angle alpha, and then the maximum gradient is calculated.  maximum gradient, gradient, directional derivative  S68 
Ordinary Differential Equations
Title and link  Description  Keywords  ID 
linear DEs  This screencast demonstrates how to solve a firstorder linear differential equation by using an integrating factor.  first order differential equations, integrating factors  S8 
Second order DEs example 2  Example of solving a secondorder linear ordinary differential equation with constant coefficients in the case where the auxiliary equation has two distinct real roots and the right hand side of the equation is an exponential function  second order differential equation, nonhomogeneous differential equation, auxiliary equation  S34 
Second order DEs example 3  Example of solving a secondorder linear ordinary differential equation with constant coefficients in the case where the auxiliary equation has complex conjugate roots and the right hand side of the equation is an exponential function  second order differential equation, nonhomogeneous differential equation, auxiliary equation  S36 
How does the integrating factor method work  Explains the integrating factor method and gives a quick proof.  differential equations, integrating factor, linear first order  S69 
First Order Linear DEs  Integrating Factors 2  This screencast shows a mediumlevel example of the solution of a first order differential equation using the integration factors method.  Linear DE, integrating factor  
Second order constant coefficient differential equations 1  This screencast first derives the auxilary equation corresponding to a second order d.e. with constant coefficients. It then shows how to find the solution of this d.e. The auxiliary equation has complex roots, and the derivation into the form e^ax(Acos(beta x)+Bsin(beta x)).  second order, differential equation, Nonhomogenous, complex roots  
First Order Linear DEs: Integrating Factors 1  This screencast shows a simple example of the solution of a first order differential equation using the integration factors method.  Linear DE, integrating factor  
First order differential equations: Separation of Variables 4  This screencast illustrates the method of separation of variables for a more advanced (and applied) example with a boundary condition: v dv/dz=g+kv^2, v(h)=0.  separation, first order, differential equation  
First order differential equations: Separation of Variables 3  This screencast illustrates the method of separation of variables for a mediumlevel first order differential equation: dv/dpsi=v(tan(psi)mu sec(psi)).  separation, first order, differential equation  
First order differential equations: Separation of Variables 2  This screencast illustrates the method of separation of variables for a mediumlevel first order differential equation: dy/dx=(2y)/(1+x). Different ways of displaying the solution are discussed.  separation, first order, differential equation  
First order differential equations: separation of variables  This screencast illustrates the method of separation of variables for a relatively simple first order differential equation  separation, first order, differential equation  
First order differential equations: solution of a Bernoulli equation  This screencast explains what is meant by a first order differential equation of Bernoulli type and solves a specific example.  Bernoulli, first order, differential equation 
Discrete Mathematics
Title and link  Description  Keywords  ID 
Karnaugh Maps with 4 variables  Example of simplifying a Boolean expression by using a 4variable Karnaugh Map  Karnaugh maps, Boolean expression, sum of products  S35 
Karnaugh Maps with 3 variables  Example of simplifying a Boolean expression by using a 3variable Karnaugh Map  Karnaugh maps, Boolean expression, sum of products  S42 
Boolean Algebra example 1  An example of using the laws of Boolean Algebra to simplify a Boolean expression, including use of De Morgan's laws.  Boolean algebra, laws of Boolean algebra, De Morgan's laws  S43 
Logic circuits  This screencast begins with explanation of 'OR', 'AND' and 'NOT' gates in a logic circuit, including what each gate looks like. An example is then given of drawing a logic circuit which contains several gates.  logic circuit, logic gates  S52 
Drawing switching circuits  This screencast starts with a discussion of what it means for switches to be connected in parallel and in series in a switching circuit and how to draw each. Two examples are then given of how to draw circuits where there are several switches with some connected in parallel and some in series in each case.  switching circuits, parallel, series  S58 
Truth table example  A demonstration of how to set up a truth table for a Boolean expression, in the context of a particular example  truth table, Boolean, switching circuit, logic  S73 
Karnaugh Map SOP and POS  With reference to a specific example, this screencast demonstrates how to use a 4variable Karnaugh map and De Morgan's laws to simplify a Boolean expression of 4 variables both into a 'sum of products' and a 'product of sums'  Boolean algebra, Karnaugh map, switching circuit, sum of products, product of sums, minimal additive form, minimal multiplicative form, SOP, POS, MAF, MMF, De Morgan's laws  S77 
Negating and Simplifying Switching Circuits  Given a diagram of a particular switching circuit, we start by identifying a Boolean function to represent it. We then find the negation of the switching circuit and finally, the simplest possible expression for it using Boolean algebra and a Karnaugh map.  Boolean algebra, Karnaugh map, switching circuit, negation, sum of products, De Morgan's laws  S78 
Karnaugh Maps and how to identify squares  In this recording, we look at how to identify '8squares', '4squares', '2squares' and '1squares' in a 4variable Karnaugh map, with reference to the different ways each of these type of groupings can appear and how to write the corresponding Boolean expression for them.  Karnaugh map, switching circuit, sum of products, 8squares, 4squares, 2squares  S79 
Designing switching circuits  An example of using a truth table to determine a Boolean expression to represent a switching circuit, given certain specified conditions as to when current will flow through the circuit  Truth table, switching circuit, Boolean algebra  S81 
2variable Karnaugh Maps  In this screencast we look at how to set up a 2variable Karnaugh map, followed by an example of simplifying a particular Boolean expression and simplifying its negation, using a 2variable Karnaugh map  Karnaugh map, negation, sum of products  S82 
Minimal spanning tree  This screencast demonstrates how the Economy Rule can be used to obtain a mininal spanning tree with application to a specific example of a cost minimisation problem for connecting four cities at minimum cost  complete graph, minimal spanning tree, discrete maths, cost minimisation problems  S14 
Binary to decimal conversion  Shows how to convert from a binary number to its corresponding decimal number. Two examples are given. 
binary, decimal, conversion  S25 
Vectors
Title and link  Description  Keywords  ID 
vector components  Explains briefly the concept behind vector components  that one vector is written as the sum of a component into a direction of another, plus the perpendicular vector. Then an example is worked through.  vectors, vector components  S32 
Equation of a plane given three points  Application of vectors to the example of finding the equation of a plane through three points by first determining two vectors on the plane, then using the cross product to determine a normal vector to the plane, then using the normal vector and any one of the three points on the plane to determine its equation  vectors, planes, geometry, cross product, vector product, normal vector  S45 
Angle between a line and a plane  Example where equation of a plane and scalar parametric equations of a line are given. To find the angle between the line and the plane, the equations of the line are converted to vector form, the angle between the line and the normal vector to the plane are found, and the angle between the line and plane is then calculated.  vectors, planes, geometry, lines, angle  S49 
Intersection of a line and a plane  Considering a line written in scalar parametric form and a plane written in form ax + by + cz = d, three examples investigate if and where line and plane intersect: in one point; the line lies on the plane; or line and plane are parallel and never meet  vectors, planes, geometry, lines, intersection, equations, unique solution, infinite solutions, no solution  S51 
Three forms of a straight line in 3D  This screencast shows the general method, together with an example, of finding the equation of a straight line in 3D in vector form, scalar parametric form and Cartesian form  vectors, straight line, vector equation, parametric equations, Cartesian equations, symmetric equations, equations without parameter  S66 
Angle between two planes  This recording starts by giving the general formula for calculating the angle between two planes by finding the angle between their normal vectors. An example is then given to illustrate this.  scalar product, angle between two vectors, planes, normal  S89 
Equation of a plane given its normal and a point on it  This recording starts by showing how the equation of a plane relates to its normal vector. An example is then done to show how this allows us to find the equation of a plane, provided we also know a point on the plane.  planes, normal, vectors  S90 
Equation of a plane given a point and a line on the plane  Given a point and a line on a plane, we find the equation of the plane.A second point on the plane is identified, and hence a vector on the plane. A normal vector to the plane is calculated using the vector product.The equation of the plane is then found.  planes, vector equation of a line, normal, cross product, vector product  S91 
Intersection of two lines in 3D  An example of first finding the vector and scalar parametric equations of two straight lines in 3D, given we know two points on each line. Gaussian Elimination then confirms that the lines intersect at a common point, whose coordinates are found. 
straight lines, lines in 3D, intersection, simultaneous equations, vectors, Guassian Elimination, unique solution, vector equation, scalar parametric equations  S93 
Matrices
Title and link  Description  Keywords  ID 
The inverse of a 3x3 matrix  the cofactor matrix method  This screencasts applies the cofactor matrix method to find the inverse of a 3x3 matrix.  cofactor matrix, adjoint matrix, determinant, inverse, 3x3  S19 
Gaussian elimination  Demonstration of the principle of solving a system of linear equations using Gaussian Elimination, with reference to a specific example with 3 equations and 3 unknowns where there is a unique solution.  linear equations, Gaussian Elimination, unique solution, matrices  S24 
Matrix multiplication  Explanation of the general method of multiplying two matrices and when matrix multiplication is defined, together with a specific example of matrix multiplication  matrix, multiplication, product  S37 
Gaussian elimination with an infinite number of solutions  An example that works through the process of Gaussian elimination for a system with three equations in three unknowns where there are an infinite number of solutions and the final answer is hence written in terms of a parameter t.  Gaussian elimination, systems of equations, 3 unknowns, infinite solutions, intersection of planes, parameter, parametric equation of a line  S38 
The determinant of a 3x3 matrix  Example of finding the determinant of a 3x3 matrix by hand, together with explanation of the general method  matrix, determinant, expanding along the first row, 3x3 matrices  S39 
Solving Equations with matrix inversion  This screencast solves a system of two equations in two unknowns by first writing the equations in matrix form and then using matrix inversion to find the solution; the general method for finding the inverse of a 2x2 matrix by hand is explained.  Inverse of a matrix, 2x2 matrix, systems of equations, 2 unknowns, solving equations, matrix inversion  S40 
Cramer's Rule for two unknowns  This screencast first gives the general form of Cramer's Rule for solving two equations in two unknowns and then covers a specific example to demonstrate the rule  Cramer's rule, systems of equations, 2 unknowns, matrix, determinant, 2x2 matrices, solving equations  S41 
Sarrus' Rule  This screencast demonstrates how Sarrus's Rule can be used to find the determinant of a 3x3 matrix. This is an alternative to the usual method of going via 2x2 subdeterminants.  Sarrus' Rule, determinant, 3x3, matrix  S46 
Matrix addition  This screencast discusses when matrix addition is defined and how it is carried out in general. This is then applied to an example involving three matrices where only two of them can be added.  Matrices, defined, matrix addition, order  S83 
Cramer's Rule for 3 unknowns  This screencast first gives the general form of Cramer's Rule for solving three equations in three unknowns and then covers a specific example to demonstrate the rule  Cramer's rule, systems of equations, 3 unknowns, matrix, determinant, 3x3 matrices, solving equations  S84 
Gaussian Elimination  no solutions  An example that works through the process of Gaussian elimination for a system with three equations in three unknowns where are no solutions.  Gaussian elimination, systems of equations, 3 unknowns, no solutions, inconsistent equations  S85 
Solving equations of form (det(Alambda_I)=0)  An example is given of solving an equation of the form det(Alambda*I)=0, where in this case A is a 2x2 matrix, I is the unit matrix and lambda is the scalar that we solve for  Unit matrix, identity matrix, determinant, scalar, solving, 2x2 matrix, eigenvalues  S87 
3D surfaces
Title and link  Description  Keywords  ID 
Sketching 3D surfaces example 1  This screencast gives an example of sketching vertical and horizontal crosssections of a surface and hence producing a final sketch of the resulting 3D surface (which is an elliptical hyperboloid of two sheets)  surfaces, ellipse, hyperbola, hyperboloid of 2 sheets, crosssections, 3 dimensional surfaces  S26 
Sketching 3D surfaces example 2  This screencast gives an example of sketching vertical and horizontal crosssections of a surface and hence producing a final sketch of the resulting 3D surface (which is a paraboloid)  surfaces, parabola, circle, paraboloid, crosssections, 3 dimensional surfaces  S27 
Sketching 3D surfaces example 3  This screencast gives an example of sketching vertical and horizontal crosssections of a surface and hence producing a final sketch of the resulting 3D surface (which is circular cones)  surfaces, straight lines, circle, cones, crosssections, 3 dimensional surfaces  S28 
Stationary point of surface  This screencast demonstrates the application of partial derivatives to finding and classifying stationary points of a 3dimensional surface, with reference to a specific example  partial derivatives, stationary points, maxima, minima, optimisation  S29 
Complex Numbers
Title and link  Description  Keywords  ID 
Complex Exponential to Cartesian Form  Three examples are given to demonstrate how to convert complex numbers written in exponential polar form into Cartesian form.  complex numbers, angle, argument, magnitude, polar form, Cartesian form  S55 
Complex Cube Roots  An example of finding the cube roots of a complex number by first converting the number from Cartesian to polar form, then hence using De Moivre's Theorem to find the roots in polar form, and then converting the roots into Cartesian form 
complex numbers, solving equations, algebra cube roots  S50 
Complex numbers division  The general principle is explained of how to divide one complex number by another when both numbers are written in Cartesian form, so that the denominator of the final answer is a real number. A specific example of using this process is then given.  algebra, complex numbers, division, fractions, conjugate  S56 
Complex polar form  General explanation is given of how a complex number in Cartesian form can be converted to polar form z=rcis(theta) and then 3 examples are given to illustrate this 
complex numbers, polar form, Cartesian form, angles, magnitude, argument  S53 
Polar form  Defines and derives the polar form of a complex number from its cartesian form 

Multiplication of complex numbers in cartesian form  Through an examples shows how to multiply complex numbers introducing the powers if i such as i^2. 

Division in cartesian form Part 1  Covers the division of complex numbers given in cartesian form introducing first the concept of the conjugate 

Division in cartesian form Part 2  Following Part 1 calculates the division of two complex numbers given in cartesian form 

Definition of a complex number  Introduces the concept of complex numbers and the cartesian form 
Curves
Title and link  Description  Keywords  ID 
Curvature and radius of curvature  An example of how to find the curvature and the radius curvature at a particular point of a curve where the equation of the curve has form y = f(x)  curves, curvature, radius of curvature, differentiation  S57 
Converting a parametric curve to Cartesian form  This screencast gives two examples of converting the parametric equations of 2D curves into Cartesian equations, and includes a discussion on the importance of considering any restrictions of the domain of the resulting Cartesian equations in each case.  parametric curves, Cartesian equation, curves, domain  S62 
Polar and Cartesian curves  This recording explains the relationship between Polar and Cartesian coordinates of a 2D curve. An example is then given of converting a polar equation into Cartesian form and an example is given of converting a Cartesian equation into polar form.  polar curves, Cartesian curves, equations, curves  S63 
Sketching polar curves  An explanation is given of the general visual interpretation of polar coordinates. An example of then given of choosing different angles, evaluating 'r' at each of these angles and then plotting and joining the resulting points by hand. 
polar curves, sketching curves, polar coordinates  S94 
Trigonometry
Title and link  Description  Keywords  ID 
non right angled triangles  This screencast gives explanation of two rules for finding unknown angles and/ or lengths of sides in any triangle: the Sine Rule and the Cosine Rule. Examples are then given to show how to apply each rule in practice.  trigonometry, sine rule, cosine rule, angles, sides, triangles  S30 
Trigonometric Ratios 1  Define the main trigonometric ratios: sine, cosine and tangent 

Trigonometric Ratios 2  Example of the solution of rightangled triangles using trigonometric ratios.  
Trigonometric ratios in all quadrants Part 1  Explains how to deal with the trig ratios of angles above 90. By using the unit circle the sign of the trig ratios in all quadrants are obtained.  
Trigonometric ratios in all quadrants Part 2  Explains how to deal with the trig ratios of angles above 90. By using the unit circle the sign of the trig ratios in all quadrants are obtained. Second part 

right angled triangles  The screencast begins by outlining the rules for finding angles in a rightangled triangle given the lengths of two of its sides. Examples are then given of applying these rules: both to finding an angle given the length of two sides in a rightangled triangle, and to finding the length of a side given the length of one of the other sides and an angle.  trigonometry, angles, sides, rightangled triangles  S31 
Pythagoras theorem  Explains the concept of a right angled triangle and introduces pythagoras theorem.  
Measuring angles  Explains the main units to measure angles: Degree and Radian 

Measuring angles Example  Example of conversion from radians to degrees and viceversa 
Number Systems
Title and link  Description  Keywords  ID 
Rules for Addition, Subtraction, Multiplication and Division  Worked examples shows the procedures for addition, subtraction, multiplication and division  Addition, Subtraction, Multiplication, Division. 
Equations
Title and link  Description  Keywords  ID 
Solving Equations Involving Fractions  Worked example shows how to solve equations involving fractions  Equations, Fractions 
Inequalities
Title and link  Description  Keywords  ID 
Introduction to Linear Inequalities  Introduction to Linear Inequalities. Definitions. Simple Examples.  Linear Inequalities, integers, real no.s, number line  
Solving Linear Inequalities  Continues on from the introduction to linear inequalities. Explains how to solve linear inequalities. Also explains why we reverse the sign if we multiply or divide by a negative number.  solving linear inequalities, multiplication/ division by negative numbers  
Solving quadratic inequalities  Continues on from the introduction to quadratic inequalities. Shows using a graph how to solve quadratic inequalities.  Quadratic Inequality, Guide Number Method,  
Introduction to quadratic inequalities  Explains what a quadratic inequality is with use of a diagram to show the difference between solving inequalities and solving equations.  Quadratic Inequality, quadratic equation  
Inequalities: Introduction  The symbols ">" and "<" are introduced. The number line is used to illustrate inequalities with positive and negative integers. This is followed with examples from fractions. 
Integration
Title and link  Description  Keywords  ID 
Integration by parts: definition  Defines the concept of integration by Parts, when it is appropriate to use IBP, and how to determine u and dv.  integration by parts, LIATE, u, dv  
Integration by parts: simple example  Applies the method of integration by parts to a simple function.  integration by parts, LIATE, u, dv, sin  
Integration by parts: integrating Ln  Applies the method of integration by parts to integrate Ln  integration by parts, LIATE, u, dv, natural log, log, ln  
Integration by parts: integrating arc tan  Applies the method of integration by parts to integrate Arc Tan  Applies the method of integration by parts to integrate Arc Tan  
Repeated integration by parts  Explains the need for repeated Integration by Parts and applies this method to a simple function.  repeated integration by parts, LIATE, u, dv, exponential, e  
Cyclical integration by parts  Applies the method of repeated integration by parts to an function containing the product of exponential and trigonometric terms resulting in a cyclical answer.  cyclical integration by parts, LIATE, u, dv, exponential, e, trigonometric, cos. cosine.  
Trigonometric integrals  Discusses trigonometric integration with simple examples.  integration, trigonometric, sin, cos  
Important integral properties  Discusses important integrand properties with simple examples.  integration, properties  
Inverse trigonometric identities (when the Xcoefficient is not one)  Discusses inverse trigonometric identities when the xcoefficient is not one.  integration, inverse trigonometric, xcoefficient  
Integration: the power rule  Definition and example of the power rule for integration. Includes exception to this power rule.  integration, power rule, exception  
Integrating exponents  Discusses how to integrate exponents with simple examples.  integration, exponents, exponential  
Inverse trigonometric identities  Discusses inverse trigonometric identities with simple examples.  integration, inverse trigonometric  
Integration by substitution: What is it?  Explains the idea behing the use of the substitution techniques and the relationship with the composition of functions. 

Integration by substitution: Power Rule Part 1  Following the previous video, the actual application of substitution is explained starting by its application to the power rule 

Integration by substitution: Power Rule Part 2  Applies the substitution for the power rule in a case where the integral needs to be rewritten by using the properties of indices before the substitution application. 

Integration by substitution: Substitution gives the integral of 1/x  Applies the substitution for the log integral in tow different examples. 

Integration by substitution: Trigonometric functions  Applies the substitution two examples one involving the integral of sin and another one involving the integral of cos 

Integration by Substitution: Exponential functions (Part 1)  Applies the substitution for the integral of an exponential of the form a^x.


Applies the substitution for the integral of an exponential of the form e^x  
Indefinite Integration as Antidifferentiation  Explains the concept of indefinite integration as the reverse of differentiation as well as the appearance of the constant of integration 