Math Academy

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.

Preliminaries

8 topics

1.1. Sets

	1.1.1.	Interior and Boundary Points
	1.1.2.	Interiors and Boundaries of Sets
	1.1.3.	Open and Closed Sets

1.2. Special Limits

1.2.1.

Limits Involving the Exponential Function

1.3. Parametric Equations

	1.3.1.	Parametric Equations of Circles
	1.3.2.	Parametric Equations of Ellipses
	1.3.3.	Parametric Equations of Parabolas
	1.3.4.	Parametric Equations of Parabolas Centered at (h,k)

Geometry & Coordinate Systems

8 topics

2.4. Quadric Surfaces and Cylinders

	2.4.1.	Spheres as Quadric Surfaces
	2.4.2.	Ellipsoids as Quadric Surfaces
	2.4.3.	Hyperboloids as Quadric Surfaces
	2.4.4.	Paraboloids as Quadric Surfaces
	2.4.5.	Elliptic Cones as Quadric Surfaces
	2.4.6.	Cylinders as Quadric Surfaces
	2.4.7.	Identifying Quadric Surfaces

2.5. Describing Two-Dimensional Space

2.5.1.

Simple, Closed, and Oriented Curves

Vector-Valued Functions

18 topics

3.6. Vector-Valued Functions

	3.6.1.	Defining Vector-Valued Functions
	3.6.2.	Limits of Vector-Valued Functions
	3.6.3.	Continuity and Differentiability of Vector-Valued Functions
	3.6.4.	Differentiation Rules for Vector-Valued Functions
	3.6.5.	Integrals of Vector-Valued Functions
	3.6.6.	Properties of Integrals of Vector-Valued Functions

3.7. Space Curves

	3.7.1.	Tangent Vectors and Tangent Lines to Curves
	3.7.2.	Unit Tangent Vectors
	3.7.3.	Principal Normal Vectors
	3.7.4.	Binormal Vectors
	3.7.5.	The Osculating Plane
	3.7.6.	The Arc Length of a Vector-Valued Function
	3.7.7.	Parameterization of a Curve by Arc Length

3.8. Curvature

	3.8.1.	Introduction to Curvature
	3.8.2.	Finding Curvature Using the Cross Product
	3.8.3.	Radius of Curvature
	3.8.4.	The Curvature of a Plane Curve
	3.8.5.	Intrinsic Coordinates

Partial Derivatives

32 topics

4.9. Multivariable Functions

	4.9.1.	Introduction to Multivariable Functions
	4.9.2.	Level Curves and Contour Plots
	4.9.3.	Level Surfaces of Multivariable Functions
	4.9.4.	Limits and Continuity of Multivariable Functions

4.10. Partial Derivatives

	4.10.1.	Introduction to Partial Derivatives
	4.10.2.	Computing Partial Derivatives Using the Rules of Differentiation
	4.10.3.	Geometric Interpretations of Partial Derivatives
	4.10.4.	Partial Differentiability of Multivariable Functions
	4.10.5.	Higher-Order Partial Derivatives
	4.10.6.	Equality of Mixed Partial Derivatives

4.11. Tangent Planes and Linear Approximations

	4.11.1.	Tangent Planes to Surfaces
	4.11.2.	Linearization of Multivariable Functions
	4.11.3.	Further Differentiability of Multivariable Functions

4.12. The Gradient Vector

	4.12.1.	The Gradient Vector
	4.12.2.	The Gradient as a Normal Vector
	4.12.3.	Tangent Lines to Level Curves
	4.12.4.	Tangent Planes to Level Surfaces
	4.12.5.	Directional Derivatives
	4.12.6.	The Multivariable Mean-Value Theorem

4.13. The Multivariable Chain Rule

	4.13.1.	The Multivariable Chain Rule
	4.13.2.	The Multivariable Chain Rule With Polar Coordinates
	4.13.3.	The Multivariable Chain Rule in Vector Form
	4.13.4.	Implicit Differentiation of Multivariable Functions
	4.13.5.	Differentials

4.14. Optimization of Scalar-Valued Functions

	4.14.1.	Defining Local and Global Extrema of Multivariable Functions
	4.14.2.	Critical Points of Multivariable Functions
	4.14.3.	The Second Partial Derivatives Test
	4.14.4.	Global Extrema of Multivariable Functions
	4.14.5.	The Hessian Matrix
	4.14.6.	Constrained Optimization
	4.14.7.	The Method of Lagrange Multipliers With One Constraint
	4.14.8.	The Method of Lagrange Multipliers With Multiple Constraints

Double Integrals

34 topics

5.15. Approximating Volumes With Riemann Sums

	5.15.1.	Double Summations
	5.15.2.	Partitions of Intervals
	5.15.3.	Calculating Double Summations Over Partitions
	5.15.4.	Approximating Volumes Using Lower Riemann Sums
	5.15.5.	Approximating Volumes Using Upper Riemann Sums
	5.15.6.	Lower Riemann Sums Over General Rectangular Partitions
	5.15.7.	Upper Riemann Sums Over General Rectangular Partitions
	5.15.8.	Defining Double Integrals Using Lower and Upper Riemann Sums

5.16. Introduction to Double Integrals

	5.16.1.	Double Integrals Over Rectangular Domains
	5.16.2.	Double Integrals Over Non-Rectangular Domains
	5.16.3.	Properties of Double Integrals

5.17. Double Integrals Over Type I and Type II Regions

	5.17.1.	Type I and II Regions in Two-Dimensional Space
	5.17.2.	Double Integrals Over Type I Regions
	5.17.3.	Double Integrals Over Type II Regions
	5.17.4.	Double Integrals Over Partitioned Regions
	5.17.5.	Changing the Order of Integration in Double Integrals

5.18. Triple Integrals

	5.18.1.	Type I, II, and III Regions in Three-Dimensional Space
	5.18.2.	Repeated Integrals in Three Dimensions
	5.18.3.	Triple Integrals Over Rectangular Domains
	5.18.4.	Triple Integrals Over Type I Regions
	5.18.5.	Triple Integrals Over Type II Regions
	5.18.6.	Triple Integrals Over Type III Regions
	5.18.7.	Calculating Volumes of Solids Using Triple Integrals
	5.18.8.	Computing Triple Integrals Using Coordinate Plane Projections
	5.18.9.	Changing the Order of Integration in Triple Integrals: Changing Projection
	5.18.10.	Changing the Order of Integration in Triple Integrals: Changing Region
	5.18.11.	Triple Integrals Over Partitioned Regions

5.19. Applications of Multiple Integrals

	5.19.1.	Calculating the Mass of a Plane Laminar
	5.19.2.	Calculating the Electric Charge of a Plane Laminar
	5.19.3.	Calculating the Center of Mass of a Plane Laminar
	5.19.4.	The Moment of Inertia of a Plane Laminar
	5.19.5.	The Radius of Gyration of a Plane Laminar
	5.19.6.	The Parallel Axis Theorem Applied to Plane Laminars
	5.19.7.	The Average Value of a Function Over a Plane Region

Change of Variables for Multiple Integrals

21 topics

6.20. Plane Transformations

	6.20.1.	Affine Transformations
	6.20.2.	The Image of an Affine Transformation
	6.20.3.	The Inverse of an Affine Transformation
	6.20.4.	The Jacobian Determinant
	6.20.5.	The Inverse Function Theorem
	6.20.6.	Nonlinear Transformations of Plane Regions
	6.20.7.	Polar Coordinate Transformations
	6.20.8.	Transformations of Regions Between Curves

6.21. Change of Variables for Double Integrals

	6.21.1.	Double Integrals in Plane Polar Coordinates
	6.21.2.	Double Integrals Between Polar Curves
	6.21.3.	Computing Areas Using a Change of Variables
	6.21.4.	Computing Double Integrals Using a Change of Variables
	6.21.5.	Computing Improper Double Integrals Using a Change of Variables

6.22. Cylindrical and Spherical Coordinates

	6.22.1.	Cylindrical Polar Coordinates
	6.22.2.	Surfaces in Cylindrical Polar Coordinates
	6.22.3.	Spherical Polar Coordinates
	6.22.4.	Surfaces in Spherical Polar Coordinates

6.23. Change of Variables for Triple Integrals

	6.23.1.	The Jacobian of a Three-Dimensional Transformation
	6.23.2.	Computing Triple Integrals Using a Change of Variables
	6.23.3.	Triple Integrals in Cylindrical Polar Coordinates
	6.23.4.	Triple Integrals in Spherical Polar Coordinates

Vector Fields

12 topics

7.24. Introduction to Vector Fields

	7.24.1.	Vector Fields
	7.24.2.	Visualizing Vector Fields
	7.24.3.	Gradient Vector Fields
	7.24.4.	Conservative Vector Fields in the Cartesian Plane
	7.24.5.	Calculating Potential Functions
	7.24.6.	Connected and Simply-Connected Regions
	7.24.7.	Conservative Vector Fields Over Simply Connected Regions

7.25. Divergence and Curl

	7.25.1.	The Divergence of a Vector Field
	7.25.2.	Properties of the Divergence Operator
	7.25.3.	The Curl of a Vector Field
	7.25.4.	Properties of the Curl Operator
	7.25.5.	Properties of the Del Operator

Line Integrals

39 topics

8.26. Line Integrals of Scalar Functions

	8.26.1.	Line Integrals of Scalar Functions
	8.26.2.	Properties of Line Integrals of Scalar Functions
	8.26.3.	Line Integrals of Scalar Functions Over Paths Expressed as Functions of X
	8.26.4.	Line Integrals of Scalar Functions Over Paths Expressed as Functions of Y

8.27. Line Integrals of Scalar Functions Over Parametric Curves

	8.27.1.	Line Integrals of Scalar Functions Over Line Segments
	8.27.2.	Line Integrals of Scalar Functions Over Circles
	8.27.3.	Line Integrals of Scalar Functions Over Ellipses
	8.27.4.	Further Properties of Line Integrals of Scalar Functions
	8.27.5.	Line Integrals of Scalar Functions Over Polar Curves

8.28. Line Integrals With Respect to X and Y

	8.28.1.	Line Integrals With Respect to X and Y
	8.28.2.	Properties of Line Integrals With Respect to X and Y
	8.28.3.	Sums of Line Integrals With Respect to X and Y Over Parametric Curves
	8.28.4.	Sums of Line Integrals With Respect to X and Y

8.29. Line Integrals of Vector-Valued Functions

	8.29.1.	Line Integrals of Vector-Valued Functions Over Parametric Curves
	8.29.2.	Line Integrals of Vector-Valued Functions Over General Curves
	8.29.3.	Interpreting Line Integrals of Vector-Valued Functions
	8.29.4.	Properties of Line Integrals of Vector-Valued Functions
	8.29.5.	The Fundamental Theorem for Line Integrals
	8.29.6.	Path Independence of Line Integrals
	8.29.7.	Conservation of Energy

8.30. Circulation and Flux

	8.30.1.	Outward-Pointing Unit Normal Vectors in 2D
	8.30.2.	Circulation
	8.30.3.	Flux in Two-Dimensional Vector Fields
	8.30.4.	Calculating Flux in Two-Dimensional Vector Fields

8.31. Green's Theorem

	8.31.1.	Green's Theorem
	8.31.2.	Green's Theorem in Polar Coordinates
	8.31.3.	Using Green's Theorem to Calculate Area
	8.31.4.	Extending Green's Theorem
	8.31.5.	Greens Theorem Applied to Regions Containing Singularities
	8.31.6.	Green's Theorem in Flux Form
	8.31.7.	Stream Functions
	8.31.8.	Source-Free Vector Fields

8.32. Applications of Line Integrals

	8.32.1.	Calculating the Mass of a Wire
	8.32.2.	Calculating the Electric Charge of a Wire
	8.32.3.	Calculating the Center of Mass of a Wire
	8.32.4.	The Moment of Inertia of a Wire
	8.32.5.	The Radius of Gyration of a Wire
	8.32.6.	The Average Value of a Function Over a Curve
	8.32.7.	Calculating the Work Done by a Force Along a Curve

Surface Integrals

25 topics

9.33. Parametric Surfaces

	9.33.1.	Parametric Surfaces
	9.33.2.	Parametrizations of Spheres, Ellipsoids and Cones
	9.33.3.	Parametrizations of Cylinders
	9.33.4.	Parametrizations of Hyperboloids and Paraboloids
	9.33.5.	Tangent Planes to Parametric Surfaces

9.34. Surface Area

	9.34.1.	Surface Areas of Revolution: Rotation About the X-Axis
	9.34.2.	Surface Areas of Revolution: Rotation About the Y-Axis
	9.34.3.	Surface Areas of Revolution for Parametric Curves
	9.34.4.	Areas of Parametric Surfaces
	9.34.5.	Surfaces of Revolution
	9.34.6.	Areas of Surfaces

9.35. Surface Integrals

	9.35.1.	Surface Integrals Over Parametric Surfaces
	9.35.2.	Surface Integrals Over Cartesian Surfaces
	9.35.3.	Flux in Three-Dimensional Vector Fields
	9.35.4.	Flux Through Closed Surfaces
	9.35.5.	Calculating Flux Through Parametric Surfaces
	9.35.6.	Calculating Flux Through Cartesian Surfaces
	9.35.7.	Calculating Flux Through Closed Surfaces
	9.35.8.	Surface Integrals in Cylindrical Polar Coordinates
	9.35.9.	Surface Integrals in Spherical Polar Coordinates

9.36. The Divergence Theorem and Stokes' Theorem

	9.36.1.	The Divergence Theorem
	9.36.2.	The Divergence Theorem With Composite Surfaces
	9.36.3.	Stokes' Theorem
	9.36.4.	Ampere’s Law
	9.36.5.	Faraday’s Law of Electromagnetic Induction

Multivariable Calculus

Outcomes

Content

2D and 3D Space

Partial Derivatives

Optimization

Double and Triple Integrals

Vector Fields

Line and Surface Integrals

Applications to Physics