Calculus 3 | Index

Calculus 3, also called multivariable calculus, extends derivatives and integrals to functions of two or three variables and to vectors in space.

The course covers vectors and the geometry of space, vector-valued functions, partial derivatives, multiple integrals and vector calculus, ending with Green’s theorem, Stokes’ theorem and the divergence theorem.

It’s for college Calculus III students and anyone taking multivariable calculus after AP Calculus BC.

What to know first: Calculus 2, especially integration techniques and parametric and polar curves, plus vectors and the dot product in Precalculus, Unit 6.

Lessons for this course are coming soon. The list below shows every planned lesson in teaching order.

Units in this course

Unit 1: Vectors and the Geometry of Space

  1. 3D Coordinate System (coming soon) — plot points, find distances and write equations of spheres in three dimensions
  2. Vectors in 3D (coming soon) — components, magnitude, unit vectors and the dot product in three dimensions
  3. Cross Product (coming soon) — find a vector perpendicular to two vectors and the area of a parallelogram
  4. Vector Projections (coming soon) — find the scalar and vector projections of one vector onto another
  5. Equations of Lines in 3D (coming soon) — write vector, parametric and symmetric equations of a line in space
  6. Equation of a Plane (coming soon) — write the equation of a plane from a point and a normal vector
  7. Quadric Surfaces (coming soon) — recognize and sketch ellipsoids, paraboloids, cones and hyperboloids

Also see: vectors, vector operations and the dot product in Precalculus, Unit 6.

Unit 2: Vector-Valued Functions

  1. Vector Functions and Space Curves (coming soon) — curves in space and the limits, derivatives and integrals of vector functions
  2. Arc Length of a Space Curve (coming soon) — find the length of a curve in three dimensions
  3. Unit Tangent and Unit Normal Vectors (coming soon) — find T and N, the directions in which a curve moves and bends
  4. Curvature (coming soon) — measure how sharply a curve bends
  5. Motion in Space (coming soon) — velocity, acceleration and the tangential and normal parts of acceleration

Also see: vector-valued functions and motion in the plane in Calculus 2, Unit 5.

Unit 3: Partial Derivatives

  1. Functions of Several Variables (coming soon) — domains, graphs, level curves and contour maps
  2. Limits of Multivariable Functions (coming soon) — find limits and show when a limit does not exist along different paths
  3. Partial Derivatives (coming soon) — differentiate with respect to one variable while holding the others fixed
  4. Tangent Planes (coming soon) — find a tangent plane and use it for linear approximation
  5. Multivariable Chain Rule (coming soon) — differentiate compositions of functions of several variables
  6. Directional Derivatives and the Gradient (coming soon) — find the rate of change in any direction and the direction of steepest increase
  7. Extrema of Functions of Two Variables (coming soon) — classify critical points with the second derivative test and find absolute extrema on closed, bounded regions
  8. Lagrange Multipliers (coming soon) — optimize a function subject to a constraint

Unit 4: Multiple Integrals

  1. Double Integrals (coming soon) — evaluate iterated integrals over rectangles
  2. Double Integrals over General Regions (coming soon) — set up and reverse the order of integration over curved regions
  3. Double Integrals in Polar Coordinates (coming soon) — integrate over disks and other round regions with polar coordinates
  4. Applications of Double Integrals (coming soon) — find mass, center of mass and surface area with double integrals
  5. Triple Integrals (coming soon) — evaluate triple integrals over solid regions
  6. Triple Integrals in Cylindrical Coordinates (coming soon) — integrate over cylinders and other round solids
  7. Triple Integrals in Spherical Coordinates (coming soon) — integrate over spheres and cones
  8. Jacobians and Change of Variables (coming soon) — change variables in multiple integrals with the Jacobian

Also see: polar coordinates in Trigonometry, Unit 9.

Unit 5: Vector Calculus

  1. Vector Fields (coming soon) — sketch and describe vector fields, including gradient fields
  2. Line Integrals (coming soon) — integrate a function or a vector field along a curve
  3. Conservative Vector Fields (coming soon) — the fundamental theorem for line integrals and how to find a potential function
  4. Green’s Theorem (coming soon) — turn a line integral around a closed curve into a double integral
  5. Curl and Divergence (coming soon) — measure the rotation and outflow of a vector field
  6. Parametric Surfaces (coming soon) — describe surfaces with two parameters and find their surface area
  7. Surface Integrals (coming soon) — integrate over a surface and find the flux of a vector field
  8. Stokes’ Theorem (coming soon) — relate a surface integral of the curl to a line integral around the boundary
  9. Divergence Theorem (coming soon) — relate the flux through a closed surface to a triple integral of the divergence

For Teachers: Standards Alignment

Lessons follow a typical college Calculus III course in multivariable calculus. There is no AP exam for this course. Two-dimensional vectors and the dot product are taught in Precalculus, Unit 6, and parametric and polar calculus in Calculus 2, Unit 5.

Previous course: Statistics Index · Next course: Linear Algebra (coming soon)