 How It Works
Courses
JOIN BETA

# Multivariable Calculus

This course is currently under construction. The target release date for this course is February.
Generalize your understanding of calculus to vector-valued functions and functions of multiple variables.

## Content

### 2D and 3D Space

• Construct equations of lines and planes.
• Classify and analyze quadric surfaces.
• Describe regions and transformations of regions in space using formal terminology.
• Convert between rectangular, cylindrical, and spherical coordinates.
• Visualize and geometrically describe the level surfaces of a function.

### Partial Derivatives

• Extend prior knowledge of single-variable derivative rules to compute partial derivatives of multivariable functions, including the chain rule and implicit differentiation.
• Use gradient notation to express the multivariable chain rule more concisely and compute directional derivatives.
• Compute differential geometry features associated with space curves and surfaces including tangents, curvature, arc length, and osculating plane.
• Use differentials and linearization to approximate values of multivariable functions.

### Optimization

• Extend prior knowledge of single-variable optimization to perform constrained optimization with multivariable functions in real-world modeling scenarios.
• Use Lagrange multipliers as an alternative method to optimize a multivariable function.

### Double and Triple Integrals

• Evaluate Riemann sums of multivariable functions and interpret them geometrically.
• Compute the integral of a multivariable function as the limit of a Riemann sum.
• Extend prior knowledge of single-variable integrals to evaluate double and triple integrals using the fundamental theorem of calculus.
• Solve advanced integrals by performing change of variables into a non-rectangular coordinate system.

### Vector Fields

• Sketch the gradient vector field for a given function.
• Determine whether a given vector field is conservative, and in the event that it is, compute a potential function for it.
• Compute and interpret divergence, curl, circulation, and flux of vector fields.

### Line and Surface Integrals

• Compute line integrals over a variety of curves.
• Use the fundamental theorem of line integrals to compute a line integral when the associated vector field is conservative.
• Understand how the fundamental theorem of line integrals implies path independence of conservative vector fields.
• Use Green’s theorem to evaluate line integrals and compute areas.
• Find parametrizations and compute surface areas of parametric surfaces.
• Evaluate surface integrals in a variety of coordinate systems.
• Use the divergence theorem and Stokes’ theorem to convert between single, double, and triple integrals when doing so eases computations.

### Applications to Physics

• Use partial derivatives to solve problems in vector mechanics.
• Use integration to calculate physical quantities for plane regions and wires and work done by a force along a curve.
1.
Preliminaries
20 topics
1.1. Sets
 1.1.1. Special Sets 1.1.2. Set-Builder Notation 1.1.3. Equivalent Sets 1.1.4. Cardinality of Sets 1.1.5. Subsets 1.1.6. The Complement of a Set 1.1.7. The Difference of Sets 1.1.8. The Cartesian Product
1.2. Finite Series
 1.2.1. Finite Linear Series 1.2.2. Finite Quadratic Series 1.2.3. Finite Cubic Series
1.3. Special Limits
 1.3.1. Special Limits Involving the Exponential Function 1.3.2. Further Special Limits Involving the Exponential Function
1.4. Transformations
 1.4.1. Affine Transformations 1.4.2. The Image of an Affine Transformation 1.4.3. The Inverse of an Affine Transformation 1.4.4. Nonlinear Transformations of Plane Regions
1.6. Parametric Equations
 1.6.1. Parametric Equations of Circles 1.6.2. Parametric Equations of Ellipses 1.6.3. Parametric Equations of Parabolas Centered at the Origin
2.
Geometry & Coordinate Systems
28 topics
2.5. Vector Geometry
 2.5.1. The Vector Equation of a Line 2.5.2. The Parametric Equations of a Line 2.5.3. Writing the Cartesian Equation of a Line from the Vector Equation 2.5.4. Finding the Vector Equation of a Plane Using the Dot Product 2.5.5. The Cartesian Equation of a Plane 2.5.6. The Parametric Equations of a Plane 2.5.7. The Intersection of Two Planes 2.5.8. The Shortest Distance Between a Plane and a Point
2.8. Describing Two-Dimensional Space
 2.8.1. Open and Closed Sets in the Plane 2.8.2. Type I and Type II Regions in Two-Dimensional Space 2.8.3. Simple, Closed, and Oriented Curves 2.8.4. Connected and Simply-Connected Regions 2.8.5. Curvilinear Coordinate Systems
2.9. Describing Three-Dimensional Space
 2.9.1. Type I, Type II and Type III Regions in Three-Dimensional Space 2.9.2. Cylindrical Polar Coordinates 2.9.3. Surfaces in Cylindrical Polar Coordinates 2.9.4. Surfaces in Cylindrical Polar Coordinates: Converting Between Cartesian and Polar Coordinates 2.9.5. Spherical Polar Coordinates 2.9.6. Surfaces in Spherical Polar Coordinates 2.9.7. Surfaces in Spherical Polar Coordinates: Converting Between Cartesian and Polar Coordinates 2.9.8. Orientable vs. Non-Orientable Surfaces
3.
Vector-Valued Functions
22 topics
3.10. Vector-Valued Functions
 3.10.1. Defining Vector-Valued Functions 3.10.2. Limits of Vector-Valued Functions 3.10.3. Derivatives of Vector-Valued Functions 3.10.4. Continuity and Differentiability of Vector-Valued Functions 3.10.5. The Rules of Differentiation for Vector-Valued Functions 3.10.6. Integrals of Vector-Valued Functions 3.10.7. Properties of Integrals of Vector-Valued Functions
3.11. Space Curves
 3.11.1. The Vector Parameterization of a Curve 3.11.2. Reversing the Direction of a Curve 3.11.3. Tangent Vectors and Tangent Lines to Curves 3.11.4. Intersecting Curves 3.11.5. Unit Tangent Vectors 3.11.6. Principal Normal Vectors 3.11.7. Binormal Vectors 3.11.8. The Osculating Plane 3.11.9. The Arc Length of a Vector-Valued Function 3.11.10. Parameterization of a Curve by Arc Length
3.12. Curvature
 3.12.1. Introduction to Curvature 3.12.2. Finding Curvature Using the Cross Product 3.12.3. Radius of Curvature 3.12.4. The Curvature of a Plane Curve 3.12.5. Intrinsic Coordinates
4.
Partial Derivatives
32 topics
4.13. Multivariable Functions
 4.13.1. Introduction to Multivariable Functions 4.13.2. Level Curves and Contour Plots 4.13.3. Level Surfaces of Multivariable Functions 4.13.4. Limits and Continuity of Multivariable Functions
4.14. Partial Derivatives
 4.14.1. Introduction to Partial Derivatives 4.14.2. Applying the Limit Definition of a Partial Derivative 4.14.3. Geometric Interpretations of Partial Derivatives 4.14.4. Partial Differentiability of Multivariable Functions 4.14.5. Higher-Order Partial Derivatives 4.14.6. Equality of Mixed Partial Derivatives
4.15. Tangent Planes and Linear Approximations
 4.15.1. Tangent Planes to Surfaces 4.15.2. Linearization of Multivariable Functions 4.15.3. Further Differentiability of Multivariable Functions
 4.16.1. The Gradient Vector 4.16.2. The Gradient as a Normal Vector 4.16.3. Tangent Lines to Level Curves 4.16.4. Tangent Planes to Level Surfaces 4.16.5. Directional Derivatives 4.16.6. The Multivariable Mean-Value Theorem
4.17. The Multivariable Chain Rule
 4.17.1. The Multivariable Chain Rule 4.17.2. The Multivariable Chain Rule With Polar Coordinates 4.17.3. The Multivariable Chain Rule in Vector Form 4.17.4. Implicit Differentiation of Multivariable Functions 4.17.5. Differentials
4.18. Optimization of Scalar-Valued Functions
 4.18.1. Defining Local and Global Extrema of Multivariable Functions 4.18.2. Calculating Critical Points of Multivariable Functions 4.18.3. The Second Partial Derivatives Test 4.18.4. Calculating Global Extrema of Multivariable Functions 4.18.5. Constrained Optimization 4.18.6. Constraints and Gradients 4.18.7. The Method of Lagrange Multipliers With One Constraint 4.18.8. The Method of Lagrange Multipliers With Multiple Constraints
5.
Double Integrals
27 topics
5.19. Approximating Volumes With Riemann Sums
 5.19.1. Double Summations 5.19.2. Computing Double Sums Using Known Results 5.19.3. Partitions of Intervals 5.19.4. Calculating Double Summations Over Partitions 5.19.5. Approximating Volumes Using Lower Riemann Sums 5.19.6. Approximating Volumes Using Upper Riemann Sums 5.19.7. Lower Riemann Sums Over General Rectangular Partitions 5.19.8. Upper Riemann Sums Over General Rectangular Partitions 5.19.9. Defining Double Integrals Using Lower and Upper Riemann Sums
5.20. Introduction to Double Integrals
 5.20.1. Evaluating Double Integrals Over Rectangular Domains 5.20.2. Defining Double Integrals Over Non-Rectangular Domains 5.20.3. Properties of Double Integrals
5.21. Double Integrals Over Type I and Type II Regions
 5.21.1. Double Integrals Over Type I Regions 5.21.2. Double Integrals Over Type II Regions 5.21.3. Double Integrals Over Partitioned Regions 5.21.4. Calculating Volumes of Solids Using Double Integrals 5.21.5. Swapping the Order of Integration in Double Integrals
5.22. Change of Variables for Double Integrals
 5.22.1. Evaluating Double Integrals Using a Change of Variables 5.22.2. Calculating Double Integrals Using Plane Polar Coordinates 5.22.3. Double Integrals of Functions Defined Between Two Polar Curves
5.23. Applications of Double Integrals
 5.23.1. Calculating the Mass of a Plane Laminar 5.23.2. Calculating the Electric Charge of a Plane Laminar 5.23.3. Calculating the Center of Mass of a Plane Laminar 5.23.4. The Moment of Inertia of a Plane Laminar 5.23.5. The Radius of Gyration of a Plane Laminar 5.23.6. The Parallel Axis Theorem Applied to Plane Laminars 5.23.7. The Average Value of a Function Over a Plane Region
6.
Triple Integrals
15 topics
6.24. Introduction to Triple Integrals
 6.24.1. Triple Summations 6.24.2. Approximating Triple Integrals Using Riemann Sums 6.24.3. Riemann Sums Over General Partitions of Rectangular Solids 6.24.4. Evaluating Triple Integrals Over Rectangular Domains 6.24.5. Defining Triple Integrals Over Non-Rectangular Domains 6.24.6. Properties of Triple Integrals
6.25. Triple Integrals Over Type I, Type II and Type III Regions
 6.25.1. Triple Integrals Over Type I Regions 6.25.2. Triple Integrals Over Type II Regions 6.25.3. Triple Integrals Over Type III Regions 6.25.4. Triple Integrals Over Partitioned Regions 6.25.5. Calculating Volumes of Solids Using Triple Integrals 6.25.6. Swapping the Order of Integration in Triple Integrals
6.26. Change of Variables for Triple Integrals
 6.26.1. Evaluating Triple Integrals Using a Change of Variables 6.26.2. Triple Integrals in Cylindrical Polar Coordinates 6.26.3. Triple Integrals in Spherical Polar Coordinates
7.
Vector Fields
19 topics
7.27. Introduction to Vector Fields
 7.27.1. Vectors in N-Dimensional Euclidean Space 7.27.2. Vector Fields 7.27.3. Visualizing Vector Fields 7.27.4. Gradient Vector Fields 7.27.5. Conservative Vector Fields Defined Over the Cartesian Plane 7.27.6. Conservative Vector Fields With Arbitrary Domains 7.27.7. Calculating a Potential Function for a Conservative Vector Field
7.28. Derivatives of Vector-Valued Functions
 7.28.1. The Jacobian of Functions of Two Variables 7.28.2. The Jacobian of Functions of Three Variables 7.28.3. Interpreting the Jacobian 7.28.4. Linearization of Vector Fields 7.28.5. The Hessian Matrix
7.29. Divergence and Curl
 7.29.1. The Divergence of a Vector Field 7.29.2. Properties of the Divergence Operator 7.29.3. The Curl of a Vector Field 7.29.4. Properties of the Curl Operator 7.29.5. Interpreting Divergence 7.29.6. Interpreting Curl 7.29.7. Properties of the Del Operator
8.
Line Integrals
38 topics
8.30. Line Integrals of Scalar Functions
 8.30.1. Line Integrals of Scalar Functions 8.30.2. Properties of Line Integrals of Scalar Functions 8.30.3. Line Integrals of Scalar Functions Over Paths Expressed as Functions of X 8.30.4. Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y
8.31. Line Integrals of Scalar Functions Over Parametric Curves
 8.31.1. Line Integrals of Scalar Functions Over Line Segments 8.31.2. Line Integrals of Scalar Functions Over Circles 8.31.3. Line Integrals of Scalar Functions Over Ellipses 8.31.4. Further Properties of Line Integrals of Scalar Functions 8.31.5. Line Integrals of Scalar Functions Over Polar Curves
8.32. Line Integrals With Respect to X and Y
 8.32.1. Line Integrals With Respect to X and Y 8.32.2. Properties of Line Integrals With Respect to X and Y 8.32.3. Sums of Line Integrals With Respect to X and Y Over Parametric Curves 8.32.4. Sums of Line Integrals With Respect to X and Y
8.33. Line Integrals of Vector-Valued Functions
 8.33.1. Line Integrals of Vector-Valued Functions Over Parametric Curves 8.33.2. Line Integrals of Vector-Valued Functions Over General Curves 8.33.3. Interpreting Line Integrals of Vector-Valued Functions 8.33.4. Properties of Line Integrals of Vector-Valued Functions 8.33.5. The Fundamental Theorem for Line Integrals 8.33.6. Path Independence of Line Integrals 8.33.7. Conservation of Energy
8.34. Circulation and Flux
 8.34.1. Outward-Pointing Unit Normal Vectors in 2D 8.34.2. Circulation 8.34.3. Flux in Two-Dimensional Vector Fields 8.34.4. Calculating Flux in Two-Dimensional Vector Fields
8.35. Green's Theorem
 8.35.1. Green's Theorem 8.35.2. Green's Theorem in Polar Coordinates 8.35.3. Using Green's Theorem to Calculate Area 8.35.4. Extending Green's Theorem 8.35.5. Greens Theorem Applied to Regions Containing Singularities 8.35.6. Green's Theorem in Flux Form 8.35.7. Stream Functions
8.36. Applications of Line Integrals
 8.36.1. Calculating the Mass of a Wire 8.36.2. Calculating the Electric Charge of a Wire 8.36.3. Calculating the Center of Mass of a Wire 8.36.4. The Moment of Inertia of a Wire 8.36.5. The Radius of Gyration of a Wire 8.36.6. The Average Value of a Function Over a Curve 8.36.7. Calculating the Work Done by a Force Along a Curve
9.
Surface Integrals
20 topics
9.37. Parametric Surfaces
 9.37.1. Parametric Surfaces 9.37.2. Parametrizations of Spheres, Ellipsoids and Cones 9.37.3. Parametrizations of Cylinders 9.37.4. Parametrizations of Hyperboloids and Paraboloids 9.37.5. Surfaces of Revolution 9.37.6. Tangent Planes to Parametric Surfaces 9.37.7. Surface Areas of Parametric Surfaces 9.37.8. Surface Areas of Special Surfaces 9.37.9. Surface Areas of Non-Parametric Surfaces
9.38. Surface Integrals
 9.38.1. Surface Integrals of Scalar Functions Over Parametric Surfaces 9.38.2. Surface Integrals of Scalar Functions Over Non-Parametric Surfaces 9.38.3. Surface Integrals of Vector-Valued Fields 9.38.4. Surface Integrals in Cylindrical Polar Coordinates 9.38.5. Surface Integrals in Spherical Polar Coordinates 9.38.6. The Flux of a Vector Field in Three Dimensions
9.39. The Divergence Theorem and Stokes' Theorem
 9.39.1. The Divergence Theorem 9.39.2. The Divergence Theorem With Composite Surfaces 9.39.3. Stokes' Theorem 9.39.4. Ampere’s Law 9.39.5. Faraday’s Law of Electromagnetic Induction