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Calculus III

Multivariable and vector calculus — partial derivatives, multiple integrals, vector fields, line and surface integrals, and the fundamental theorems (Green's, Stokes', Divergence) that generalize the FTC to higher dimensions.

5 modules·86 concepts·50 practice problems·~45 hours
Practice Calc IIIBrowse Modules

Prerequisites

Calculus IILinear Algebra

Exam Relevance

University Exams1 exam
University Calculus III100% of exam

Module Breakdown

1.Vectors & Geometry in 3D

17 concepts·14 problems

Three-dimensional coordinate geometry, vectors in R³, dot and cross products, equations of lines and planes, quadric surfaces, and cylindrical and spherical coordinate systems.

17 concepts covered
2d Vectors And Components ReviewPolar Coordinates Review3d Coordinate SystemVectors In R3 Components And MagnitudeUnit Vectors And Direction CosinesDot Product In 3dProjections And Work Via Dot ProductCross Product Definition And ComputationCross Product Geometric Interpretation And PropertiesScalar Triple ProductEquations Of Lines In 3dEquations Of PlanesDistances Point To Plane And Line To LineQuadric SurfacesCylindrical CoordinatesSpherical CoordinatesConverting Between Coordinate Systems

2.Vector-Valued Functions

13 concepts·6 problems

Functions mapping scalars to vectors — parametric curves in space, velocity and acceleration vectors, curvature, and the Frenet-Serret (TNB) frame.

13 concepts covered
Parametric Curves ReviewVector Valued Functions Definition And GraphsLimits And Continuity Of Vector FunctionsDerivatives Of Vector FunctionsDifferentiation Rules For Vector FunctionsIntegrals Of Vector FunctionsArc Length In 3dArc Length ParameterizationCurvatureUnit Tangent And Unit Normal VectorsBinormal Vector And Tnb FrameTangential And Normal Components Of AccelerationVelocity And Acceleration In 3d

3.Partial Derivatives

21 concepts·10 problems

Differentiation of multivariable functions: partial derivatives, the gradient, directional derivatives, tangent planes, the chain rule, and optimization with Lagrange multipliers.

21 concepts covered
Single Variable Differentiation ReviewFunctions Of Two Variables Graphs And Level CurvesFunctions Of Three Variables And Level SurfacesLimits And Continuity For Multivariable FunctionsPartial Derivatives Definition And ComputationHigher Order Partial DerivativesClairauts Theorem Equality Of Mixed PartialsTangent Planes To SurfacesLinear Approximation In Several VariablesTotal DifferentialMultivariable Chain RuleImplicit Differentiation For Multivariable FunctionsDirectional DerivativeGradient VectorGradient And Level Curves SurfacesTangent Planes Via The GradientCritical Points Of Multivariable FunctionsSecond Derivative Test DiscriminantAbsolute Extrema On Closed RegionsLagrange Multipliers One ConstraintLagrange Multipliers Two Constraints

4.Multiple Integrals

17 concepts·12 problems

Double and triple integrals: area, volume, mass, center of mass, moments of inertia. Fubini's theorem, change of order, and coordinate transformations (polar, cylindrical, spherical, Jacobian).

17 concepts covered
Single Variable Integration Techniques ReviewPolar And Cylindrical Coordinates ReviewDouble Integrals Over Rectangles Fubinis TheoremDouble Integrals Over General RegionsSwitching The Order Of IntegrationDouble Integrals In Polar CoordinatesApplications Area And Volume Via Double IntegralsMass And Center Of Mass 2dMoments Of Inertia 2dTriple Integrals In Cartesian CoordinatesTriple Integrals In Cylindrical CoordinatesTriple Integrals In Spherical CoordinatesMass And Center Of Mass 3dMoments Of Inertia 3dChange Of Variables And The JacobianSurface Area Via Double IntegralsChoosing Integration Order And Coordinate System

5.Vector Calculus

18 concepts·12 problems

Vector fields, line integrals, surface integrals, and the three great theorems — Green's, Stokes', and the Divergence Theorem — that generalize the Fundamental Theorem of Calculus.

18 concepts covered
Work As A Line Integral ReviewVector Fields Definition And VisualizationLine Integrals Of Scalar FunctionsLine Integrals Of Vector Fields Work IntegralsFundamental Theorem For Line IntegralsConservative Vector Fields And Potential FunctionsIndependence Of PathGreens Theorem Circulation FormGreens Theorem Flux FormApplications Of Greens TheoremCurl Of A Vector FieldDivergence Of A Vector FieldPhysical Interpretation Of Curl And DivergenceParametric Surfaces In 3dSurface Integrals Of Scalar FunctionsOriented Surfaces And Surface Integrals Of Vector Fields FluxStokes TheoremDivergence Theorem Gausss Theorem

Reference Textbooks

  • Stewart — Multivariable Calculus
  • Thomas — Calculus: Multivariable

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