Sign up
Multivariable Calculus · Lessons

Multivariable Calculus lessons

10 sections · 63 skills · work through the full outline skill by skill — read the long-form lesson or flip through the slide-style version.
Long-form lessonsSlide-style lessons

Multivariable Calculus

Practice 63 skills across this outline.

Recommended next
Triple Integrals: Spherical
Multiple Integrals · Triple Integrals
Long-form lesson
Weighting distribution
18 categories
Vector Fields
10%
Multivariable Functions
10%
Surface Integrals
10%
Line Integrals
10%
Integral Theorems
10%
Optimization in Several Variables · Unconstrained Optimization
6%

Skills

The comprehensive taxonomy of multivariable calculus — drill into a card to work its skills.

Multiple Integrals · Triple Integrals

4 core skills3% total weight
0/4 completed
4 skills to mastery
1
Triple Integrals: SphericalRecommended
Long-form
2
Triple Integrals: CylindricalAvailable
Long-form
3
Mass, Center of Mass & MomentsAvailable
Long-form
4
Triple Integrals: CartesianAvailable
Long-form

Multiple Integrals · Double Integrals in Polar Coordinates

3 core skills2% total weight
0/3 completed
3 skills to mastery
1
Jacobian in Polar CoordinatesAvailable
Long-form
2
Area in Polar CoordinatesAvailable
Long-form
3
Double Integrals in PolarAvailable
Long-form

Multiple Integrals · Double Integrals

4 core skills2% total weight
0/4 completed
4 skills to mastery
1
Changing Integration OrderAvailable
Long-form
2
Area & Average Value: Double IntegralsAvailable
Long-form
3
Double Integrals: General RegionsAvailable
Long-form
4
Iterated IntegralsAvailable
Long-form

Multiple Integrals · Change of Variables and Jacobians

2 core skills2% total weight
0/2 completed
2 skills to mastery
1
Change of VariablesAvailable
Long-form
2
Jacobian DeterminantsAvailable
Long-form

Vector Fields

4 core skills10% total weight
0/4 completed
4 skills to mastery
1
DivergenceAvailable
Long-form
2
CurlAvailable
Long-form
3
Conservative Fields & PotentialsAvailable
Long-form
4
Vector Fields & Field LinesAvailable
Long-form

Multivariable Functions

3 core skills10% total weight
0/3 completed
3 skills to mastery
1
Surfaces & Contour MapsAvailable
Long-form
2
Limits & Continuity: MultivariableAvailable
Long-form
3
Domains & Level SetsAvailable
Long-form

Surface Integrals

4 core skills10% total weight
0/4 completed
4 skills to mastery
1
Surface Integrals: Scalar FieldsAvailable
Long-form
2
Flux Across SurfacesAvailable
Long-form
3
Surface Parameterization & AreaAvailable
Long-form
4
Orientation & Normal VectorsAvailable
Long-form

Line Integrals

4 core skills10% total weight
0/4 completed
4 skills to mastery
1
Line Integrals: Vector FieldsAvailable
Long-form
2
Parametrizing CurvesAvailable
Long-form
3
Line Integrals: Scalar FieldsAvailable
Long-form
4
Fundamental Theorem for Line IntegralsAvailable
Long-form

Integral Theorems

4 core skills10% total weight
0/4 completed
4 skills to mastery
1
Choosing TheoremsAvailable
Long-form
2
Stokes' TheoremAvailable
Long-form
3
Divergence TheoremAvailable
Long-form
4
Green's TheoremAvailable
Long-form

Optimization in Several Variables · Unconstrained Optimization

3 core skills6% total weight
0/3 completed
3 skills to mastery
1
Finding Critical PointsAvailable
Long-form
2
Second Derivative Test & HessianAvailable
Long-form
3
Constrained OptimizationAvailable
Long-form

Optimization in Several Variables · Constrained Optimization

2 core skills4% total weight
0/2 completed
2 skills to mastery
1
Lagrange MultipliersAvailable
Long-form
2
Lagrange Multiplier InterpretationAvailable
Long-form

Partial Derivatives · Computing Partial Derivatives

5 core skills6% total weight
0/5 completed
5 skills to mastery
1
Clairaut's TheoremAvailable
Long-form
2
Partial DerivativesAvailable
Long-form
3
Differentials & Error EstimationAvailable
Long-form
4
Tangent Planes & Linear ApproximationAvailable
Long-form
5
Multivariable Chain RuleAvailable
Long-form

Partial Derivatives · Directional Derivatives and Gradient

4 core skills4% total weight
0/4 completed
4 skills to mastery
1
Gradient: Steepest Ascent/DescentAvailable
Long-form
2
Gradient VectorAvailable
Long-form
3
Level Curves & Gradient OrthogonalityAvailable
Long-form
4
Directional DerivativesAvailable
Long-form

Problem-Solving & Modeling Tools · Setup and Visualization

3 core skills3% total weight
0/3 completed
3 skills to mastery
1
Sketching Regions & BoundsAvailable
Long-form
2
Choosing ParameterizationsAvailable
Long-form
3
Using SymmetryAvailable
Long-form

Problem-Solving & Modeling Tools · Computation Workflow

3 core skills6% total weight
0/3 completed
3 skills to mastery
1
Units, Orientation & SignsAvailable
Long-form
2
Common PitfallsAvailable
Long-form
3
Selecting Coordinate MethodsAvailable
Long-form

Vectors and Geometry in 3D · Lines, Planes, and Surfaces

4 core skills2% total weight
0/4 completed
4 skills to mastery
1
3D Line EquationsAvailable
Long-form
2
Plane Equations & NormalsAvailable
Long-form
3
Quadric SurfacesAvailable
Long-form
4
Distances & Angles: Lines/PlanesAvailable
Long-form

Vectors and Geometry in 3D · Coordinate Systems

3 core skills4% total weight
0/3 completed
3 skills to mastery
1
Cartesian to CylindricalAvailable
Long-form
2
Cartesian to SphericalAvailable
Long-form
3
Choosing Coordinate SystemsAvailable
Long-form

Vectors and Geometry in 3D · Vectors and Dot/Cross Products

4 core skills4% total weight
0/4 completed
4 skills to mastery
1
Cross ProductAvailable
Long-form
2
Dot ProductAvailable
Long-form
3
Scalar Triple ProductAvailable
Long-form
4
Vector Components & MagnitudeAvailable
Long-form