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Calculus IIIOnline College Course
Multivariable calculus — vectors, partial derivatives, multiple integrals, and the great integral theorems.
What You'll Learn
A full semester of calculus iii — completed at your own pace, with no deadlines or fixed class times.
What You'll Be Able to Do
- Perform vector operations in 2D and 3D space — including dot product, cross product, and projections — and apply them to geometric problems involving lines, planes, and quadric surfaces
- Analyze vector-valued functions using calculus to compute derivatives, integrals, arc length, curvature, and model motion in space
- Compute and interpret partial derivatives, gradients, and directional derivatives for multivariable functions, and apply the chain rule and linear approximation
- Solve constrained and unconstrained optimization problems for multivariable functions using critical point analysis and Lagrange multipliers
- Evaluate double and triple integrals in Cartesian, polar, cylindrical, and spherical coordinates, and apply them to compute area, volume, mass, and center of mass
- Compute line integrals of scalar and vector fields, apply the Fundamental Theorem for Line Integrals, and use Green's Theorem to relate line and double integrals
- Evaluate surface integrals over parametric and explicit surfaces, compute flux integrals, and apply Stokes' Theorem and the Divergence Theorem
Major Course Topics
Course chapters and lessons
Calculus III has 7 chapters across 46 instructional hours, as listed in the course syllabus. Expand any chapter to see the lessons inside it.
- Chapters
- 7
- Lessons
- 38
- Instructional hours
- 46 hrs
- Semester hours
- 4
Chapter 1
Vectors & the Geometry of Space
- 1.1 Three-Dimensional Coordinate Systems plotting points in 3D, distance formula, equation of a sphere
- 1.2 Vectors vector representation, addition, scalar multiplication, unit vectors, standard basis vectors
- 1.3 The Dot Product computation, geometric interpretation, angle between vectors, projections, scalar and vector projection
- 1.4 The Cross Product determinant formula, geometric interpretation, area of a parallelogram, triple scalar product
- 1.5 Equations of Lines and Planes parametric, symmetric, and vector equations of lines; scalar and vector equations of planes; parallel & perpendicular planes; distance from a point to a plane
- 1.6 Cylinders and Quadric Surfaces identifying and sketching cylinders, ellipsoids, paraboloids, cones, hyperboloids
Chapter 2
Vector-Valued Functions & Space Curves
- 2.1 Vector Functions and Space Curves definition, domain, limits, continuity, sketching space curves
- 2.2 Derivatives and Integrals of Vector Functions derivative rules, tangent vector, smooth curves, definite integrals
- 2.3 Arc Length and Curvature arc length formula, curvature κ, unit tangent T, unit normal N, binormal B (TNB frame), torsion
- 2.4 Motion in Space: Velocity and Acceleration position, velocity, speed, acceleration, projectile motion, circular motion
Chapter 3
Partial Derivatives
- 3.1 Functions of Several Variables domain, range, graphs, level curves, level surfaces
- 3.2 Limits and Continuity computing limits along paths, proving limits do not exist, continuity for multivariable functions
- 3.3 Partial Derivatives notation, computation, higher-order partials, Clairaut's Theorem
- 3.4 Tangent Planes and Linear Approximation equation of tangent plane, linearization L(x,y), differentials
- 3.5 The Chain Rule chain rule for compositions of multivariable functions, implicit differentiation
- 3.6 Directional Derivatives and the Gradient Vector directional derivative, gradient ∇f, steepest ascent, tangent planes to level surfaces
Chapter 4
Optimization of Multivariable Functions
- 4.1 Maximum and Minimum Values local and absolute extrema, critical points, second derivative test (discriminant D)
- 4.2 Absolute Extrema on Closed Bounded Regions finding absolute max/min by testing critical points and boundary
- 4.3 Lagrange Multipliers method of Lagrange multipliers for one and two constraints, applied optimization problems
Chapter 5
Multiple Integrals
- 5.1 Double Integrals over Rectangles Riemann sums, definition, iterated integrals, Fubini's Theorem
- 5.2 Double Integrals over General Regions type I and type II regions, changing order of integration
- 5.3 Double Integrals in Polar Coordinates setting up and evaluating in polar form
- 5.4 Applications of Double Integrals area, volume, mass, center of mass, moments of inertia, probability
- 5.5 Surface Area computing surface area using double integrals
- 5.6 Triple Integrals in Cartesian Coordinates definition, Fubini's Theorem, volume, mass
- 5.7 Triple Integrals in Cylindrical Coordinates setup and evaluation
- 5.8 Triple Integrals in Spherical Coordinates setup and evaluation, Jacobian
- 5.9 Change of Variables in Multiple Integrals Jacobian of a transformation, change of variables formula
Chapter 6
Vector Calculus
- 6.1 Vector Fields definition, visualization, conservative (gradient) fields
- 6.2 Line Integrals line integrals of scalar functions and vector fields (work), over piecewise-smooth curves
- 6.3 The Fundamental Theorem for Line Integrals path independence, conservative fields, potential functions
- 6.4 Green's Theorem statement, proof sketch, applications to area and circulation
- 6.5 Curl and Divergence definitions, geometric interpretation, ∇ operator, test for conservative fields
Chapter 7
Surface Integrals & Integral Theorems
- 7.1 Parametric Surfaces and Their Areas parametrization of surfaces, tangent planes, surface area formula
- 7.2 Surface Integrals surface integrals of scalar functions, orientation of surfaces
- 7.3 Flux Integrals surface integrals of vector fields, flux through a surface
- 7.4 Stokes' Theorem statement and application, relating surface integrals to line integrals
- 7.5 The Divergence Theorem statement and application, relating flux through closed surface to triple integral of divergence
These chapters come straight from the official course syllabus, which also sets out the graded assignment and the proctored final exam.
Course Details
| Course | Calculus 3 (MATH 365) |
|---|---|
| Format | 100% online |
| Learning Style | Self-paced — no deadlines or class times |
| Credits | 4 College Credits |
| Lab | Not required |
| Prerequisites | Calculus 2 |
| Assessment | 25% Attendance · 25% Quizzes · 25% Assignments · 25% Final Exam |
| Estimated Completion | Self-paced · no deadlines |
| Credit Recommendation | NCCRS Credit Recommended |
| Transfer Information | Transferable via the credit recommendation above — your registrar can confirm your school's policy |
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Master calculus iii and build a foundation for your degree — and your future.
Multivariable calculus — vectors, partial derivatives, multiple integrals, and the great integral theorems.
🎯 Why It Matters
- Calculus III is required for engineering, physics, mathematics, and many computer science degrees
- Multivariable methods underpin machine learning, optimization, and graphics
- Vector calculus is foundational to electromagnetism and fluid dynamics
- Completing the calculus sequence unlocks upper-division STEM coursework
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🚀 How This Sets You Up
- Complete 7 chapters from vectors through the integral theorems
- Quiz per subchapter, one assignment, 100-question final exam
- 46 contact hours, fully self-paced
- Completes the standard three-semester calculus sequence
- Prepares you for differential equations and linear algebra
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A 57-second look at Calculus III.
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Everything You Need to Succeed
Video-Based Lessons
HD video lectures broken into short, focused modules. Watch at your own pace — pause, rewind, replay any time.
Quizzes & Final Exam
Evaluate your understanding after every section with quizzes and a comprehensive final exam that mirrors real college assessment.
Structured College-Level Course
Structured lessons covering every topic — designed as a standard Calculus III college-level course.
Official Transcript
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