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Calculus 3

Multivariable calculus and vector analysis

218Modules

7Units

Vectors and 3D Geometry

Vectors, dot products, cross products, lines, planes, and surfaces

33 modules

Forces in SpaceINTRO

Measuring AlignmentINTRO

Surface GalleryINTRO

Creating Perpendicular VectorsINTRO

Vector FundamentalsLESSON

Unit Vectors and DirectionLESSON

Dot Product PropertiesLESSON

Dot Product ApplicationsPRACTICE

Cross Product Definition and PropertiesLESSON

Cross Product MasteryPRACTICE

Triple Products and VolumeLESSON

Equations of Lines in 3DLESSON

Distance CalculationsLESSON

Quadric SurfacesLESSON

Cylindrical and Spherical SurfacesLESSON

Work and Energy CalculatorREAL WORLD

Flight Path NavigationREAL WORLD

Sketching z = x² + y²EXAMPLE

Finding Line Through P(1,2,3) and Q(4,-1,2)EXAMPLE

Finding Resultant ForceEXAMPLE

Computing u × v for u = <1, 2, 3> and v = <2, -1, 1>EXAMPLE

Finding Angle Between VectorsEXAMPLE

Projection ExplorerACTIVITY

Plane Equation GeneratorACTIVITY

Torque and Moment SimulatorACTIVITY

Surface Equation BuilderACTIVITY

Vector Operations LabACTIVITY

Surface IdentificationPRACTICE

Vector ArithmeticPRACTICE

Lines and Planes AnalyzerPRACTICE

Vectors and 3D Geometry SummarySUMMARY

3D Geometry Reference SheetFORMULA SHEET

Section 1 AssessmentQUIZ

Vector Functions

Vector-valued functions, curves, arc length, and motion in space

27 modules

Roller Coaster DesignINTRO

Measuring Curved PathsINTRO

Introduction to Vector FunctionsLESSON

Limits and ContinuityLESSON

Derivatives of Vector FunctionsLESSON

Integration of Vector FunctionsLESSON

Arc Length of Space CurvesLESSON

Space Curve TracerACTIVITY

Arc Length and Curvature CalculationsPRACTICE

Finding Arc Length of r(t) = <3t, 4sin t, 4cos t> from t=0 to t=2πEXAMPLE

The Curvature FormulaLESSON

Velocity and AccelerationLESSON

Tangential and Normal ComponentsLESSON

Velocity from PositionREAL WORLD

Projectile Motion LabREAL WORLD

Analyzing the Helix r(t) = <cos t, sin t, t>EXAMPLE

Analyzing Circular Motion r(t) = <R cos(ωt), R sin(ωt), h>EXAMPLE

Finding r'(t) for r(t) = <t², sin(2t), e^t>EXAMPLE

TNB Frame ConstructorACTIVITY

Curvature ExplorerACTIVITY

Motion AnalyzerACTIVITY

Vector Function EvaluationPRACTICE

Space Motion AnalysisPRACTICE

Vector Calculus OperationsPRACTICE

Vector Functions SummarySUMMARY

Space Curves Reference SheetFORMULA SHEET

Section 2 AssessmentQUIZ

Partial Derivatives

Partial derivatives, gradients, directional derivatives, and optimization

39 modules

Maximum Rate of ChangeINTRO

Composition ChainsINTRO

Best Flat ApproximationINTRO

Rates in Specific DirectionsINTRO

Approaching from All DirectionsINTRO

Temperature MapsINTRO

Multivariable FunctionsLESSON

Level Sets and ContoursLESSON

Limits in Multiple VariablesLESSON

Continuity for Multivariable FunctionsLESSON

Limits and Continuity AnalysisPRACTICE

Defining Partial DerivativesLESSON

Higher-Order PartialsLESSON

Tangent Planes to SurfacesLESSON

Tangent Planes and ApproximationsPRACTICE

Finding Tangent Plane to z = x²+xy+y² at (1,2,7)EXAMPLE

Differentiability and DifferentialsLESSON

Chain Rule for Several VariablesLESSON

Chain Rule Diagram BuilderACTIVITY

Chain Rule ApplicationsPRACTICE

Implicit DifferentiationLESSON

Directional DerivativesLESSON

Gradient and Directional Derivative ProblemsPRACTICE

Finding Directional Derivative at P(2,1) in Direction <3,4>EXAMPLE

Properties of the GradientLESSON

Computing dz/dt for z=x²y where x=cos t, y=sin tEXAMPLE

Finding Partials of f(x,y) = x³y² - 2xy + y³EXAMPLE

Investigating lim_(x,y)→(0,0) xy/(x²+y²)EXAMPLE

Analyzing f(x,y) = x² - y²EXAMPLE

Gradient Field ExplorerACTIVITY

Linearization LaboratoryACTIVITY

Partial Derivative CalculatorACTIVITY

Limit Path ExplorerACTIVITY

Surface VisualizerACTIVITY

Partial Derivative ComputationsPRACTICE

Function Evaluation and DomainsPRACTICE

Partial Derivatives SummarySUMMARY

Partial Derivatives Reference SheetFORMULA SHEET

Section 3 AssessmentQUIZ

Multiple Integrals

Double and triple integrals in various coordinate systems

39 modules

Natural Coordinate SystemsINTRO

3D Mass DistributionINTRO

Circular SymmetryINTRO

Volume Under a SurfaceINTRO

Double Integral DefinitionLESSON

Properties of Double IntegralsLESSON

Integration over General RegionsLESSON

Changing Order of IntegrationLESSON

Double Integrals in PolarLESSON

Area and Volume in PolarLESSON

Surface AreaLESSON

Mass and Center of MassLESSON

Triple Integral DefinitionLESSON

Triple Integrals over General RegionsLESSON

Cylindrical CoordinatesLESSON

Spherical CoordinatesLESSON

Center of BalanceREAL WORLD

Irregular TerritoriesREAL WORLD

Volume of Cone z = √(x²+y²) under z = 4EXAMPLE

Evaluating ∭_E xyz dV over Box [0,1]×[0,2]×[0,3]EXAMPLE

Finding COM of Triangle with Density ρ(x,y) = x+yEXAMPLE

Finding Volume over Circle x²+y² ≤ 4 under f(x,y) = 4-x²-y²EXAMPLE

Integrating x+2y over Triangle with Vertices (0,0), (2,0), (0,3)EXAMPLE

Evaluating ∬_R (x²+y) dA over R=[0,2]×[1,3]EXAMPLE

Coordinate System SelectorACTIVITY

Triple Integral VisualizerACTIVITY

Moment of Inertia CalculatorACTIVITY

Polar Integration DesignerACTIVITY

Region Descriptor ToolACTIVITY

Riemann Sum VisualizerACTIVITY

Applications in Alternative CoordinatesPRACTICE

Triple Integration PracticePRACTICE

Applications PracticePRACTICE

Polar Integration PracticePRACTICE

General Region IntegrationPRACTICE

Double Integrals over RectanglesPRACTICE

Multiple Integration SummarySUMMARY

Multiple Integration Reference SheetFORMULA SHEET

Section 4 AssessmentQUIZ

Vector Calculus

Line integrals, surface integrals, and the fundamental theorems

40 modules

Generalizing Green's TheoremINTRO

Circulation and FluxINTRO

Path IndependenceINTRO

Flow PatternsINTRO

Introduction to Vector FieldsLESSON

Line Integrals of Scalar FunctionsLESSON

Line Integrals of Vector FieldsLESSON

Conservative Vector FieldsLESSON

Fundamental Theorem of Line IntegralsLESSON

Statement of Green's TheoremLESSON

Applications of Green's TheoremLESSON

Parametric SurfacesLESSON

Surface Integral of Scalar FunctionLESSON

Surface Integrals of Vector FieldsLESSON

Stokes' TheoremLESSON

The Divergence TheoremLESSON

Divergence Theorem LaboratoryACTIVITY

Flux Through SurfacesREAL WORLD

Conservation of EnergyREAL WORLD

Work Along a PathREAL WORLD

Verifying Stokes' for F = <-y, x, z> over HemisphereEXAMPLE

Finding Mass of Hemisphere z = √(4-x²-y²) with ρ = zEXAMPLE

Verifying Green's Theorem for F = <y², x> over CircleEXAMPLE

Using FTC for ∫_C F·dr where F = <2xy, x²+1>EXAMPLE

Evaluating ∫_C xy ds where C is Quarter CircleEXAMPLE

Sketching F(x,y) = <-y, x>EXAMPLE

Flux VisualizerACTIVITY

Green's Theorem CalculatorACTIVITY

Potential Function FinderACTIVITY

Path Integral CalculatorACTIVITY

Vector Field LaboratoryACTIVITY

Applications of Major TheoremsPRACTICE

Surface Integral PracticePRACTICE

Green's Theorem ApplicationsPRACTICE

Path Independence PracticePRACTICE

Line Integral ApplicationsPRACTICE

Vector Field AnalysisPRACTICE

Vector Calculus SummarySUMMARY

Vector Calculus Reference SheetFORMULA SHEET

Section 5 AssessmentQUIZ

Advanced Topics

Additional advanced topics in multivariable calculus

33 modules

When Exact Solutions FailINTRO

Local Polynomial ApproximationINTRO

Finding Peaks and ValleysINTRO

Critical Points and ClassificationLESSON

Global ExtremaLESSON

Finding Extrema of f(x,y) = x³ - 3xy + y³EXAMPLE

Method of Lagrange MultipliersLESSON

Lagrange Multiplier VisualizerACTIVITY

Economic InterpretationLESSON

Taylor Polynomials for f(x,y)LESSON

Finding Quadratic Taylor Polynomial of f(x,y) = e^x cos y at (0,0)EXAMPLE

Applications to Error AnalysisLESSON

Partial Differential EquationsLESSON

Electromagnetic FieldsLESSON

Numerical IntegrationLESSON

Gradient DescentLESSON

Heat DistributionREAL WORLD

Constrained OptimizationREAL WORLD

Approximating ∬_D e^(x²+y²) dA over Unit DiskEXAMPLE

Solving Laplace Equation ∇²u = 0EXAMPLE

Maximizing f(x,y) = xy subject to x² + y² = 8EXAMPLE

Numerical Methods LaboratoryACTIVITY

Fluid Flow SimulatorACTIVITY

Taylor Series BuilderACTIVITY

Optimization ExplorerACTIVITY

Numerical Methods PracticePRACTICE

Physics ApplicationsPRACTICE

Taylor Series PracticePRACTICE

Constrained Optimization ApplicationsPRACTICE

Optimization PracticePRACTICE

Advanced Topics SummarySUMMARY

Advanced Topics Reference SheetFORMULA SHEET

Section 6 AssessmentQUIZ

conclusion

Course Conclusion

Review and synthesis of course material

7 modules

Real-World Applications PortfolioACTIVITY

3D Calculus GalleryACTIVITY

Journey from 1D to 3DSUMMARY

Problem-Solving Strategy SynthesizerSUMMARY

Calculus 3 Concept MapSUMMARY

Master Formula CompendiumFORMULA SHEET

Comprehensive Final ExamQUIZ