The role of the local tangent plane in approximating function changes due to small changes in the independent variables will be discussed, followed by the refinement of this technique through the use of Taylor series representations of the function. Tools for analysing the convergence properties of such power series will be presented, permitting sensible conclusions to be drawn from expressions involving infinite sums. Basic concepts of differential calculus, including limits, continuity and differentiability are reviewed and generalized to functions of more than one variable.
Final topics deal with aspects of integration in three dimensions.
Parametric expressions are considered that often lead to the most efficient representations for space curves, simplifying the calculation of such integral quantities as arc length. Integration over two-dimensional domains on the other hand leads to volumes. These integrals may usually be evaluated as a sequence of one-dimensional integrals. The material in this paper is fundamental to much of Mathematics and Statistics, as well as being widely used in all the physical sciences, Engineering, Surveying, Economics and Finance.