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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 second-order 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 first-order linear differential equation by using an integrating factor. first order differential equations, integrating factors S8
Second order DEs example 2 Example of solving a second-order 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, non-homogeneous differential equation, auxiliary equation S34
Second order DEs example 3 Example of solving a second-order 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, non-homogeneous 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 medium-level 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, Non-homogenous, 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 medium-level 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 medium-level first order differential equation: dy/dx=(2-y)/(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 4-variable Karnaugh Map Karnaugh maps, Boolean expression, sum of products S35
Karnaugh Maps with 3 variables Example of simplifying a Boolean expression by using a 3-variable 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 4-variable 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 '8-squares', '4-squares', '2-squares' and '1-squares' in a 4-variable 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, 8-squares, 4-squares, 2-squares 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
2-variable Karnaugh Maps In this screencast we look at how to set up a 2-variable Karnaugh map, followed by an example of simplifying a particular Boolean expression and simplifying its negation, using a 2-variable 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 co-factor matrix method This screencasts applies the co-factor matrix method to find the inverse of a 3x3 matrix. co-factor 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 sub-determinants. 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(A-lambda_I)=0) An example is given of solving an equation of the form det(A-lambda*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 cross-sections 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, cross-sections, 3 dimensional surfaces S26
Sketching 3D surfaces example 2 This screencast gives an example of sketching vertical and horizontal cross-sections of a surface and hence producing a final sketch of the resulting 3D surface (which is a paraboloid) surfaces, parabola, circle, paraboloid, cross-sections, 3 dimensional surfaces S27
Sketching 3D surfaces example 3 This screencast gives an example of sketching vertical and horizontal cross-sections of a surface and hence producing a final sketch of the resulting 3D surface (which is circular cones) surfaces, straight lines, circle, cones, cross-sections, 3 dimensional surfaces S28
Stationary point of surface This screencast demonstrates the application of partial derivatives to finding and classifying stationary points of a 3-dimensional 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 right-angled 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 right-angled 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 right-angled triangle, and to finding the length of a side given the length of one of the other sides and an angle. trigonometry, angles, sides, right-angled 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 X-coefficient is not one) Discusses inverse trigonometric identities when the x-coefficient is not one. integration, inverse trigonometric, x-coefficient  
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