Section 1 Preface
Calculus III was primarily devoted to extending the derivative to multivariable scalar functions. This course starts with the parallel program of extending the integral to multivariable scalar functions. Unlike the derivative, the definition of the multivariable (definite) integral is relativley similar to the single variable case. The challenge of integration is in the geometry of the subsets of \(\RR^n\) involved in the integration. The course proceeds to a number of applications of multivariable integration.
The second half of the course covers the material known as cector calculus. This is the calculus of multivariable vector functions: functions with multivariable outputs as well as inputs. We investigate several derivative and integral extensions for these functions.
Since the extension of the derivative to multivariable functions is multi-faceted, we loose the simple idea of an antiderivative. However, in the context of vector calculus, we can produce several new theorems which resemble the fundamental theorem of calculus, even without a simple anti-derivative. The fruit of this labour are the famous Gauss, Green and Stokes theorems.
The final section of the course is a brief introduction to differential geometry. We introduce the ideas of manifolds and differential forms (and we finally get a reasonable definition of the differential ‘dx’ used in integrals). We develop this material far enough to show the deep connections underlying the fundamental theorem of calculus and the Gauss, Green and Stokes theorems.