Appendix A Vector Calculus in Two Dimensions

The purpose of this appendix is to review the basics of vector calculus in the two dimensions. Most, if not all, this material should be familiar to the student who has taken a basic course in multivariable calculus, but it is worth collecting together the necessary results. We will assume you are familiar with the basics of partial derivatives, including the equality of mixed partical (assuming they are continuous), the chain rule, implcit differentiation. In addition, some familiarity with multiple integrals is assumed, although we will review the highlights. Proofs can be found in most vector calculus text, including [9, 168, 171]. We begin with a discussion of plane curves and domains. Many physical quantities, including force and velocity, are determined by vector fields, and we review the basic concepts. The key differential operators in planar vector calculus are the gradient and divergence operations, along with the Jacobian matrix for maps from R to itself. There are three basic types of line integrals: integrals with respect to arc length, for computing lengths of curves, masses of wires, center of mass, etc., ordinary line integrals of vector fields for computing work and fluid circulation, and flux line integrals for computing flux of fluids and forces. Next, we review the basics of double integrals of scalar functions over plane domains. Line and double integrals are connected by the justly famous Green’s theorem, which is the two-dimensional version of the fundamental theorem of calculus. The integration by parts argument required to characterize the adjoint of a partial differential operator rests on the closely allied Green’s formula.