Video of the lecture.

Began with some explanation of the second online homework.

Then considered a business maths problem whose solution is obtained by integrating a differential 2-form over a planar region.

Then explained the rules for differentiating a differential 1-form w in order to obtain a 2-form dw.

I'm taking the view that: (i) differential n-forms are things represented by symbols; (ii) I'll not worry too much in lectures about what they; (ii) the online homeworks are devoted to explaining what n-forms are; (iv) I'll be very precise in lectures about what is meant by the integral of an n-form over an oriented n-dimensional region; (v) in order to calculate integrals of n-forms we'll need to know the rules for manipulating and differentiating n-forms; (vi) these manipulation rules are forced on us by our understanding of an integral of an n-form and by the desire to have Stoke's formula hold.

So at the end of the module we'll be able to integrate over n-dimensional regions using Elie Cartan's language of differential forms, a language that is perfect for:

providing one simple and unified explanation of a range of what, in other approaches, would appear to be ad hoc notions and results (such as curl, div, grad, the Fundamental Theorem of Calculus, Green's Theorem, Gauss' Theorem, ...)

providing a generalization of basic ideas of calculus to n-dimensions,

giving a concise description of electromagnetism.