We discuss integrals with a linear function of sin or cos in the denominator of the integrand. We address the question of existence then exhibit a substitution which allows for use of the residue theorem to evaluate the integral.

Part 1 of 3. We show how to swap the order of integration in a double integral depending on x and y. We use the same examples as for the region of integration for double integrals. We start with constant integration limits and progress to variable limits.

Part 2 of 3. Continuation from part one, looking at increasingly complicated examples of reversal of integration order.

Introduces the del operator and gives brief discussion of how it is used as a gradient, divergence and curl, then in the formation of the Laplacian.

Looks at how linear factors to odd and even powers can help in sketching polynomial functions that can be factorised into all linear factors.

We solve some basic exercises to decide on the linear dependence, or not, of some sets of vectors in 2 and 3 dimensions. These simple cases can be done by inspection without Gaussian elimination.

Example of getting the polar form of a complex number if the cartesian form is known(number placed in the first quadrant).

Knowing the Laplace transform of a function f(t) allows us also to calculate the transform of the integral of f(t). We show the simplest method of doing this here.

We show how to find the eigenvectors for the 3 by 3 matrix whose eigenvalues were calculated in a separate presentation.

We use the same 2 by 2 matrix whose eigenvalues were found in another MathsCast (Eigenvalues of a 2 by 2 matrix) and find the assocIated eigenvectors.