In this lecture we introduce the core concept of the vector derivative. This is the natural extension of the gradient operator to geometric algebra setting, where it inherits the properties of a grade-1 vector. The separate inner and outer products account for the divergence and curl operations in 3D, and extend both to higher dimensions and grade.
The Cauchy-Riemann equations have a simple expression in terms of the vector derivative, and the geometric algebra form generalises to higher dimensions, recovering the Maxwell and Dirac equations in spacetime. The fundamental theorem of geometric calculus provides the link between surface integrals and volume integrals of a differentiated quantity. We end by showing how the fundamental theorem includes the famous Cauchy integral formula and can extend the concept to higher dimensions.