APPLIED MATHEMATICS
OVERVIEW
Vectors 
and scalars in 4-D space become the headline attractions for Calculus III. From cartesian coordinates in space, to vectors in space, terms, like the norm of a vector, a unit vector , and accompanying laws of addition, subtraction and scalar multipliers are learned. Coulomb's law is demonstrated applying the principle of superposition. Three key areas of vectors are the dot product , the cross product and triple product . Later, lines and planes in vectoral space are explored within limit and continuity theory. Derivatives and intergrals of both lines, planes and closed surfaces in 3D vectoral space are explained, eventually applying tangential planes and curvature with respect to Kepler's Laws of Motion .
Functions of several variables and their accompanying applications of limit and continuity theories lead us to our first exposure of Partial Derivatives . Applications learned in Calculus I are again used with partial derivatives. Directional derivatives are introduced leading us to a crucial part of Calculus—the gradient . Tangential planes of approximations to Lagrange multipliers follow quickly on to double and triple integrals within cartesian and polar coordinate systems.
Surface area applications of non-linear volumes are determined, to direct applications in physics beyond their geometries and towards their mass and charge properties. Triple integrations in spherical coordinates aide in finding the physical applications of moment and centers of mass of solid body regions. Finally, changes in variables or their partial derivatives within multiple integrals are explored leading us finally to calculus of vector fields. The curl , or vector field and it's many applications including electromagnetics is introduced. We re-apply line integrals, now within vector fields, to determine Work and it's force curve. Gradients and the fundamental theorem of line integrals expose us to the Law of Conservation of Energy , ultimately exposing us to Green's Theorem [1a] Surface integrals and integrals over oriented surfaces prepares us for the three dimensional version of Green's Theorem known as Stoke's Theorem . Boundary equations for Green's application 2D surfaces and Stoke's application of 3D surfaces ultimately ends with The Divergence Theorem for Vector Fields.[1]
