Calculus 3: Multivariable Calculus, Talked Through Step by Step

A taste of a lesson on Lagrange Multipliers

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The 35 topics in Calculus 3

Each topic builds on the ones before it. The tutor checks what you already know and starts there.

1. Vectors in 2D and 3D

   Represent points and vectors in the plane and in space using coordinates; perform vector addition, scalar multiplication, and compute magnitude and unit vectors. Understand vectors as representing displacement, direction, and magnitude, and how to visualize them geometrically in 3D coordinate systems.

2. Dot Product

   Compute the dot product of two vectors algebraically and understand its geometric meaning via the angle between vectors. Use the dot product to test orthogonality, find projections of one vector onto another, and compute work done by a force.

   Builds on Vectors in 2D and 3D

3. Cross Product

   Compute the cross product of two 3D vectors using the determinant formula, and understand its geometric meaning as a vector orthogonal to both inputs with magnitude equal to the area of the parallelogram they span. Use the cross product and scalar triple product to compute areas and volumes.

   Builds on Vectors in 2D and 3D

4. Lines and Planes in Space

   Write parametric and vector equations of lines in 3D using a point and direction vector; find the equation of a plane using a point and normal vector. Determine intersections, distances, and relationships (parallel, perpendicular, skew) between lines and planes.

   Builds on Dot Product, Cross Product

5. Quadric Surfaces and Cylinders

   Recognize and sketch standard quadric surfaces (ellipsoids, paraboloids, hyperboloids, cones) and cylinders from their equations by analyzing cross-sectional traces. Understand how coefficients affect the shape and orientation of the surface.

   Builds on Vectors in 2D and 3D

6. Vector-Valued Functions and Space Curves

   Define a vector-valued function of one parameter as tracing a curve in space; compute derivatives and integrals of vector functions component-wise. Understand the derivative as a tangent vector and use it to find tangent lines to curves.

   Builds on Vectors in 2D and 3D

7. Arc Length and Curvature

   Compute the arc length of a space curve via integration of speed, and reparametrize a curve by arc length. Define curvature using the derivative of the unit tangent vector, and compute the unit tangent, normal, and binormal vectors (TNB frame).

   Builds on Vector-Valued Functions and Space Curves

8. Motion in Space: Velocity and Acceleration

   Model position, velocity, and acceleration as vector-valued functions of time; decompose acceleration into tangential and normal components. Apply these ideas to problems involving projectile motion and circular motion in space.

   Builds on Arc Length and Curvature

9. Functions of Several Variables

   Understand functions of two or more variables, their domains, and how to represent them using graphs, level curves, and level surfaces. Interpret level curves as contour maps and use them to understand the shape of a surface.

   Builds on Vectors in 2D and 3D

10. Limits and Continuity in Several Variables

    Understand what it means for a limit of a multivariable function to exist, including the requirement that the limit be the same along every path of approach. Use path-dependence to show limits do not exist, and understand continuity in terms of limits matching function values.

    Builds on Functions of Several Variables

11. Partial Derivatives

    Compute first and higher-order partial derivatives of multivariable functions by differentiating with respect to one variable while holding others fixed. Understand partial derivatives as instantaneous rates of change in a coordinate direction and interpret them as slopes of tangent lines to traces.

    Builds on Limits and Continuity in Several Variables

12. Tangent Planes and Linear Approximation

    Find the equation of the tangent plane to a surface at a point using partial derivatives, and use it to linearly approximate function values near that point. Understand the tangent plane as the best linear approximation to a differentiable surface, connecting to the concept of differentiability.

    Builds on Partial Derivatives

13. Multivariable Chain Rule

    Apply the chain rule to compute derivatives of composite functions where intermediate variables depend on one or more other variables, using tree diagrams to organize dependencies. Use the chain rule for implicit differentiation of multivariable equations.

    Builds on Tangent Planes and Linear Approximation

14. Directional Derivatives and the Gradient

    Define the directional derivative as the rate of change of a function in a specified direction, and compute it using the gradient vector. Understand the gradient as pointing in the direction of steepest ascent, with magnitude equal to the maximum rate of increase, and its relationship to level curves as being perpendicular to them.

    Builds on Multivariable Chain Rule

15. Local Maxima, Minima, and Saddle Points

    Find critical points of a multivariable function by setting partial derivatives to zero, and classify them as local maxima, minima, or saddle points using the second derivative (discriminant) test. Understand geometrically what a saddle point represents.

    Builds on Directional Derivatives and the Gradient

16. Lagrange Multipliers

    Find extrema of a function subject to one or more equality constraints using the method of Lagrange multipliers, setting the gradient of the objective proportional to the gradient of the constraint. Understand the geometric idea of gradients being parallel at a constrained extremum, and apply the method to optimization problems.

    Builds on Local Maxima, Minima, and Saddle Points

17. Double Integrals over Rectangular Regions

    Define the double integral as the limit of Riemann sums giving signed volume under a surface, and evaluate it using iterated integrals over rectangular regions via Fubini's theorem. Understand the geometric meaning of the double integral as volume.

    Builds on Functions of Several Variables

18. Double Integrals over General Regions

    Set up and evaluate double integrals over regions bounded by curves, choosing the order of integration and correctly determining variable bounds of type I and type II regions. Switch the order of integration when needed to simplify computation.

    Builds on Double Integrals over Rectangular Regions

19. Double Integrals in Polar Coordinates

    Convert double integrals to polar coordinates, including the correct area element r dr dθ, to evaluate integrals over circular or annular regions. Recognize when polar coordinates simplify a problem based on the region's or integrand's symmetry.

    Builds on Double Integrals over General Regions

20. Applications of Double Integrals

    Use double integrals to compute quantities such as area, mass of a lamina with variable density, center of mass, and moments of inertia. Set up the correct integrand for each physical quantity based on density functions.

    Builds on Double Integrals in Polar Coordinates

21. Triple Integrals in Rectangular Coordinates

    Set up and evaluate triple integrals over solid regions in rectangular coordinates as iterated integrals, determining the bounds for each variable based on the geometry of the solid. Understand the triple integral as representing total accumulation of a density function over a volume.

    Builds on Applications of Double Integrals

22. Triple Integrals in Cylindrical Coordinates

    Convert triple integrals to cylindrical coordinates using the volume element r dr dθ dz, and use them to evaluate integrals over solids with circular symmetry around an axis.

    Builds on Triple Integrals in Rectangular Coordinates

23. Triple Integrals in Spherical Coordinates

    Convert triple integrals to spherical coordinates using the volume element ρ² sin(φ) dρ dφ dθ, and use them to evaluate integrals over solids with spherical symmetry, such as spheres and cones centered at the origin.

    Builds on Triple Integrals in Cylindrical Coordinates

24. Change of Variables and the Jacobian

    Generalize substitution to multiple integrals using a transformation between coordinate systems and the Jacobian determinant to correctly scale the area or volume element. Apply this to simplify integrals via non-standard substitutions beyond polar, cylindrical, and spherical coordinates.

    Builds on Triple Integrals in Spherical Coordinates

25. Vector Fields

    Understand a vector field as an assignment of a vector to each point in space, visualize vector fields by sketching representative vectors, and recognize examples like gravitational and velocity fields. Distinguish conservative vector fields as gradients of a scalar potential function.

    Builds on Directional Derivatives and the Gradient

26. Line Integrals of Scalar Functions

    Define and compute line integrals of scalar functions over a curve, parametrizing the curve and integrating with respect to arc length. Apply this to compute quantities like the mass of a wire with variable density.

    Builds on Vector-Valued Functions and Space Curves, Double Integrals over Rectangular Regions

27. Line Integrals of Vector Fields

    Define and compute the line integral of a vector field along a curve, representing work done by a force field along a path. Understand orientation-dependence of the integral and compute it via parametrization.

    Builds on Vector Fields, Line Integrals of Scalar Functions

28. Fundamental Theorem for Line Integrals

    Understand that the line integral of a conservative vector field depends only on the endpoints of the path, not the path itself, via a fundamental theorem analogous to single-variable calculus using the potential function. Use this to test for path-independence and to find potential functions for conservative fields.

    Builds on Line Integrals of Vector Fields

29. Green's Theorem

    State and apply Green's Theorem to relate a line integral around a simple closed curve in the plane to a double integral over the enclosed region, converting between circulation/flux forms and area integrals. Use Green's theorem to compute areas and simplify line integral computations.

    Builds on Double Integrals over General Regions, Fundamental Theorem for Line Integrals

30. Curl and Divergence

    Compute the curl and divergence of a vector field using partial derivatives, and interpret curl as measuring local rotation and divergence as measuring the outward flux density (source/sink strength) at a point. Use curl to test whether a vector field is conservative.

    Builds on Green's Theorem

31. Parametric Surfaces and Surface Area

    Represent surfaces parametrically using two parameters, compute the resulting normal vector via the cross product of partial derivatives, and use it to find the surface area. Apply this to surfaces of revolution and graphs of functions of two variables.

    Builds on Cross Product, Multivariable Chain Rule

32. Surface Integrals of Scalar Functions

    Define and compute surface integrals of scalar functions over a parametrized surface, using the surface area element derived from the cross product of partial derivatives. Apply this to compute quantities like the mass of a curved shell with variable density.

    Builds on Parametric Surfaces and Surface Area

33. Surface Integrals of Vector Fields (Flux)

    Define and compute the flux of a vector field through an oriented surface as a surface integral, representing quantities like fluid flow rate through a membrane. Determine surface orientation and its effect on the sign of the flux integral.

    Builds on Curl and Divergence, Surface Integrals of Scalar Functions

34. Stokes' Theorem

    State and apply Stokes' Theorem to relate the surface integral of the curl of a vector field to the line integral of the field around the boundary curve, generalizing Green's Theorem to surfaces in 3D. Use it to simplify computations of circulation or flux of curl.

    Builds on Surface Integrals of Vector Fields (Flux)

35. Divergence Theorem

    State and apply the Divergence Theorem (Gauss's Theorem) to relate the flux of a vector field through a closed surface to the triple integral of its divergence over the enclosed solid. Use it to simplify flux computations and understand its physical interpretation in terms of sources and sinks.

    Builds on Triple Integrals in Rectangular Coordinates, Stokes' Theorem

Practice questions in Calculus 3

A few of the kinds of questions you will work through, with answers.