We write the directional derivative on a surface z=z(x,y) in terms of the grad operator acting on z as a dot product with a unit vector in the direction required.

Multivariate chain rule example (MathsCasts)

May 17th, 2016 7m 25s

Example of using the multivariate chain rule where in this case z=f(x,t) and x is a function of t. In this example the expression relating x and t is an implicit expression so implicit differentiation is required in finding dx/dt for that part

A scheme to make integration by parts easier - Part 1 (MathsCasts)

May 27th, 2014 9m 44s

Integration by parts using the formula is briefly revisited, then we look at how to code use of the formula into a visual process of adding terms that might be easier to remember.

The complex Fourier series of e^t (MathsCasts)

Mar 22nd, 2014 8m 4s

The complex Fourier coefficients are calculated for the function e^t with period 2 pi.

Differentation using the inverse function rule (MathsCasts)

Mar 22nd, 2014 6m 53s

In this recording use the inverse function rule for differentiation to find the derivate of the inverse sine of x. This example includes discussion of the importance of looking at the graph of such a function, to see whether it is increasing or

Matrix multiplication using index notation (MathsCasts)

Mar 22nd, 2014 8m 18s

We show how to use index notation and sum over row and column indices to perform matrix multiplication. The Einstein summation convention is introduced.

Bonus results from Fourier series - Part 3 (Using Parseval's theorem) (MathsCasts)

Mar 18th, 2014 7m 10s

Parseval's theorem is used to achieve a bonus result for the sum of the series of reciprocal fourth powers of odd integers.

Bonus results from Fourier series - Part 2 (MathsCasts)

Mar 17th, 2014 4m 45s

Further use of Fourier series to find the sum of a series involving alternating signed reciprocals of odd integers.

Integration by substitution: What do we do with the integration limits? (MathsCasts)

Mar 17th, 2014 8m 20s

We describe how to deal with integration limits when substitution is performed on a definite integral.

Bonus results from Fourier series - Part 1 (MathsCasts)

Mar 16th, 2014 8m 34s

Starting with a known Fourier series, we derive results of sumes of series involving reciprocals of powers of the whole numbers.

Irrotational vector fields, potential function and path integral - Part 2 (MathsCasts)

Jan 8th, 2014 10m 31s

A vector field is introduced, it is shown to be irrotational then its potential is identified and the field is integrated along a specified path.

Irrotational vector fields, potential function and path integral - Part 1 (MathsCasts)

Jan 7th, 2014 11m 1s

We explain the concept of an irrotational vector field and potential for such a field, then show how the potential is used to calculate a path integral of the vector filed given end-points for the path. A specified vector field shown to be irro

Irrotational vector fields and the potential: Part 3 another example (MathsCasts)

Jan 6th, 2014 7m 9s

Another irrotational vector field is introduced, and its potential is identified. The field is then integrated along a specified path.

Introduction to Cartesian tensors - Part 1 The Kronecker delta (MathsCasts)

Jan 5th, 2014 11m 22s

We introduce the Kronecker delta and identify it as just another way of writing the unit matrix.

Closed contour line integral - Part 2 Using Green's theorem in the plane (MathsCasts)

Jan 4th, 2014 6m 46s

Evaluates the same line integral as was done in part 1, but using Green's theorem in the plane

Taylor Series: Logarithmic Functions (MathsCasts)

Dec 9th, 2013 3m 55s

Calculates the Taylor series for the natural logarithmic function about the point x = 1.

Taylor Series: Definition (MathsCasts)

Dec 8th, 2013 3m 4s

Firstly introduces the notion of approximating a function via a polynomial before defining the Taylor and MacLaurin series.

Maclaurin Series: Trigonometric Functions (MathsCasts)

Nov 18th, 2013 5m 23s

Calculates the MacLaurin series for the trigonometric function sine x. Comments on the pattern of the series.

Maclaurin Series: Exponential Functions (MathsCasts)

Nov 17th, 2013 4m 46s

Calculates the MacLaurin series for the exponential function. Also discusses the MacLaurin series for e^2x derived from the corresponding series for e^x.

Maclaurin Series: Composite Functions (MathsCasts)

Nov 16th, 2013 5m 12s

Calculates the MacLaurin series for a composite function e^{sinx}. Highlights the fact that this series must be a polynomial.

Partial fractions: Multiplicity of a root (MathsCasts)

Oct 24th, 2013 5m 40s

Explains the partial fraction expansion procedure when the denominator has repeated linear roots, and applies this technique to a simple example. Then applies this expansion to conduct the integration. Uses substitution in the resulting integra

Reciprocal Trigonometric Integrals (MathsCasts)

Oct 24th, 2013 5m 24s

Discusses reciprocal trigonometric identities and investigates numerous examples, some of which involve reading the differentiation tables.

Partial fractions: Mixed roots - Part 1 (MathsCasts)

Oct 23rd, 2013 6m 29s

Explains the partial fraction expansion procedure when the denominator has a mixture of linear roots and repeated linear roots, and applies this technique to a simple example.

Partial fractions: Mixed roots - Part 2 (MathsCasts)

Oct 23rd, 2013 2m 20s

Uses the expansion from Part 1 to conduct the integration. Uses substitution in the resulting integrals.

Partial fractions: Linear roots - Part 2 (MathsCasts)

Oct 13th, 2013 4m 7s

Firstly double checks the expansion from Part 1. Then shows the use of partial fractions to conduct the integration. Uses substitution in the resulting integrals.