### Course Description and Outline

 Multivariable Calculus is a continuation of Calculus II, but is mostly concerned with generalizing the main concepts from Calculus I, such as differentiation and integration, to higher dimensions. Topics will cover include vector algebra and geometry; parametric curves in 3-dimensional space; differentiation and integration of scalar functions of several variables with applications to maximum-minimum problems and finding areas and volumes; differentiation and integration of vector fields; and Green’s, Stokes’, and Divergence Theorems (see below for a more detailed list of topics). At the end of the course, you should have a good understanding of how calculus of functions of two or more variables works and be able to apply this knowledge to a variety of problems arising in math and physics, such as motion in space, approximation of functions, optimization, density, mass, etc. The more general objective of this course is to continue to provide you with a deeper understanding and working knowledge of calculus, while in the process strengthening your analytical skills, increasing your ability to communicate mathematics symbolically and orally, making you comfortable with reading and understanding mathematics on your own, and developing an appreciation for calculus as one of the greatest intellectual developments in history. Here is a more detailed list of topics we will cover:Vector algebra and geometryvector algebra and interpretation in spacedot product, cross product, orthogonality, direction angles, projectionQuadric surfacesequations and visualizations of cylinders, paraboloids, hyperboloids, etc.Calculus of one-parameter vector functions in spaceparametric equations of lines and curvesvector equations of lines and curveslimits and continuityderivatives of vector functionstangent vectors and linesinteraction of the derivative with dot and cross productsintegrals of vector functionsarc length and curvaturenormal and binormal vectors (time permitting)velocity and accelerationCalculus of functions of two or more variablessurfaces and level curveslimits and continuitydifferentiationpartial derivativesClairaut’s Theoremtangent planeslinear approximation (time permitting)differentialsThe Chain Ruleimplicit differentiation (time permitting)directional derivatives, the gradient, and interpretationsoptimizationLagrange multipliers (time permitting)integrationdouble integrals over rectangles and their propertiesintegral as volumeaverage valueiterated integrals and Fubini’s Theoremdouble integrals over general regionsdouble integrals in polar coordinatesdensity, mass, center of masssurface areatriple integrals over boxes and general regionstriple integrals in cylindrical and spherical coordinatesgeneral change of variables and the Jacobian (time permitting)Vector calculusvector fields in the plane and the three-spacegradient fieldsline integralsline integral of vector fields and work along a curveThe Fundamental Theorem for Line Integralsconservative vector fields and independence of pathGreen’s Theoremcurl and divergence of a vector field and interpretationsparametric equations of surfacesgrid curvessurfaces of revolutiontangent planessurface area (time permitting)surface integralssurface integrals (flux) of vector fieldsStokes’ TheoremDivergence Theorem (time permitting)