bythreedu · Math Tracks · Curriculum v1.0 · 2026

Vectors, surfaces,
and calculus in
three dimensions.

A complete Calculus III course covering vectors and 3D geometry, vector functions, partial derivatives, multiple integrals, and vector calculus through Stokes\' and the Divergence Theorem. Built for students in STEM who need multivariable calculus as a foundation for differential equations, physics, and engineering.

Total Lessons18at your pace
Units6progressive phases
Assignments6+problem sets
DestinationDiff Eq
Ready
00Approach
Course Philosophy

Geometry Is the Foundation

Multivariable calculus lives in three-dimensional space. Before any formula is introduced, this course builds geometric intuition for vectors, surfaces, and regions. Every theorem — from the gradient to Stokes — is understood geometrically before it is applied computationally.

Coordinate Systems Are Tools

Rectangular, polar, cylindrical, spherical — each coordinate system exists to make a specific class of problem tractable. This course teaches you to recognize which system is right before you start, not after you've fought through an integral that didn't need to be hard.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–3
Vectors and 3D Space
3 lessons · Vectors · Dot and cross products · Lines and planes
dot productcross productvector equations
+
02
Unit 2 · Weeks 4–5
Vector Functions and Space Curves
2 lessons · Vector derivatives · Arc length · Curvature
vector derivativesarc lengthcurvature
+
03
Unit 3 · Weeks 6–8
Multivariable Functions and Partial Derivatives
3 lessons · Partial derivatives · Gradient · Directional derivatives
partial derivativesgradientdirectional derivative
+
04
Unit 4 · Weeks 9–11
Optimization and the Chain Rule
3 lessons · Chain rule · Implicit differentiation · Optimization
multivariable chain ruleLagrange multiplierssecond derivative test
+
05
Unit 5 · Weeks 12–14
Multiple Integrals
3 lessons · Double integrals · Triple integrals · Change of variables
double integralspolar coordinateschange of variables
+
06
Unit 6 · Weeks 15–18
Vector Calculus and Capstone
4 lessons + capstone · Line integrals · Green's theorem · Stokes and divergence
line integralsGreen's theoremStokes' theorem
+
02Capstone
Simulated Final Exam

The capstone is a full-length Calc III final exam simulation. Every missed problem becomes a lesson — not just the right answer, but the reasoning that should have driven each step. The goal is to walk into Differential Equations without a single gap.

○
Gap Review
Two weakest units revisited before the final exam simulation.
○
Timed Final Exam
Full-length, independent, same conditions as a real exam.
○
Written Corrections
Every missed problem corrected with reasoning — not just the right answer.
○
Differential Equations Readiness Review
A final session on what Differential Equations will demand from day one.
03Investment
Pricing Options
Location
per Session
$125
1.5 hrs · online only
Session length
  • —Flexible scheduling
  • —1-on-1 focused attention
  • —No long-term commitment

Best for trying the curriculum before committing to a plan.

Popular
per Month
$500
4 sessions · 1.5 hrs/session
Frequency
Session length
  • —Session notes after every lesson
  • —Async DM support between sessions
  • —Priority scheduling

Adjust frequency and session length above — price updates automatically.

Per Semester
$2,000
16 sessions · 1.5 hrs/session
Term length
Frequency
Session length
  • —All 6 units — full course arc
  • —Session notes after every lesson
  • —Capstone simulated final + written corrections
  • —Differential Equations readiness review at completion

The full Calculus III arc — from vectors to the Divergence Theorem — in one commitment.

📍
In-person available · NYC onlyMonthly and semester plans can be done in-person. Select “In-person · NYC” above to see adjusted pricing.
04Contact
Let's Talk

Finished Calc II and ready to go multivariable? One session is enough to see where you are and plan the arc forward.

What do I need coming in?+
Are lessons online or in-person?+
Can I switch plans?+
What if I need to reschedule?+