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.

Duration~18weeks
Total Lessons18at your pace
Units6progressive phases
Assignments6+problem sets
DestinationDiff Eq Ready
00Approach
Course Philosophy

Geometry Drives Everything

Multivariable calculus is inseparable from geometry. The gradient, the Jacobian, the curl, the divergence — each has a geometric meaning that makes it understandable. This course teaches the geometry alongside the computation so the formulas make sense.

The Big Theorems Are the Point

Green's, Stokes', and the Divergence Theorem aren't just computation tools — they are the most beautiful results in undergraduate mathematics. This course builds toward them deliberately so that when they arrive, students understand why they are true.

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
+
L1
Vectors in 2D and 3D
Vector arithmetic, magnitude, unit vectors. Position vectors. The geometric meaning of vector addition and scalar multiplication.
L2
Dot Product and Cross Product
Dot product as projection — and as the cosine formula. Cross product as the area of a parallelogram and the normal to a plane. Right-hand rule.
L3
Lines and Planes in 3D
Parametric and symmetric equations of lines. Equation of a plane from a normal vector and a point. Distances from points to lines and planes.
Assignment 1
Find the angle between 5 pairs of vectors. Find the equation of 3 planes from given conditions. Compute 5 cross products and verify using the dot product test.
You work fluently in three-dimensional space using vectors.
02
Unit 2 · Weeks 4–5
Vector Functions and Space Curves
2 lessons · Vector derivatives · Arc length · Curvature
vector derivativesarc lengthcurvature
+
L4
Vector Functions
Limits, derivatives, and integrals of vector functions. Tangent vectors, velocity, and acceleration in 3D. Smooth curves.
L5
Arc Length, Curvature, and TNB Frames
Arc length of a space curve. Curvature as the rate of change of the unit tangent. The TNB (Frenet) frame as the natural coordinate system for a curve.
Assignment 2
Parameterize 3 curves, compute their tangent vectors, and find arc length over a given interval. Find the curvature of 2 curves.
You analyze space curves using vector calculus.
03
Unit 3 · Weeks 6–8
Multivariable Functions and Partial Derivatives
3 lessons · Partial derivatives · Gradient · Directional derivatives
partial derivativesgradientdirectional derivative
+
L6
Functions of Several Variables
Domains, ranges, and level curves. Reading a contour map. Limits in two variables — why they are harder than in one variable.
L7
Partial Derivatives
Computing and interpreting partial derivatives. Higher-order partials. Clairaut's theorem. The geometric meaning of a partial derivative as a slope.
L8
Gradient and Directional Derivatives
The gradient as the direction of steepest ascent. Directional derivatives as the rate of change in any direction. The gradient is perpendicular to level curves — proved, not asserted.
Assignment 3
Compute all first and second-order partial derivatives for 5 functions. Find the gradient and 3 directional derivatives for each. Interpret one in plain English.
You differentiate multivariable functions and understand what each derivative means geometrically.
04
Unit 4 · Weeks 9–11
Optimization and the Chain Rule in Multiple Variables
3 lessons · Chain rule · Implicit differentiation · Optimization
multivariable chain ruleLagrange multiplierssecond derivative test
+
L9
Multivariable Chain Rule
The chain rule for compositions of several variables. Tree diagrams for the dependencies. Implicit differentiation in two variables.
L10
Critical Points and the Second Derivative Test
Finding critical points. The discriminant D = f_xx f_yy - f_xy² and what it tells you. Saddle points vs. local extrema.
L11
Optimization with Constraints: Lagrange Multipliers
Lagrange multipliers derived from the condition that the gradient of f is parallel to the gradient of g. Economic and geometric interpretations.
Assignment 4
Find and classify all critical points for 4 functions. Solve 3 Lagrange multiplier problems. Write the geometric interpretation of one result.
You optimize multivariable functions with and without constraints.
05
Unit 5 · Weeks 12–14
Multiple Integrals
3 lessons · Double integrals · Triple integrals · Change of variables
double integralspolar coordinateschange of variables
+
L12
Double Integrals
Iterated integrals over rectangular and general regions. Reversing the order of integration. Average value of a function over a region.
L13
Double Integrals in Polar Coordinates
When polar coordinates make the integral tractable. The Jacobian factor r. Finding area, volume, and mass using polar double integrals.
L14
Triple Integrals and Change of Variables
Triple integrals in rectangular, cylindrical, and spherical coordinates. The Jacobian for general changes of variables. When to use each coordinate system.
Assignment 5
Evaluate 8 double integrals — 4 rectangular, 4 polar. Set up 3 triple integrals in the most appropriate coordinate system.
You evaluate multiple integrals in the coordinate system that makes each tractable.
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
+
L15
Line Integrals
Scalar and vector line integrals. Work done by a force field. Path independence and conservative fields. The gradient theorem.
L16
Green's Theorem
Relating a line integral around a closed curve to a double integral over the enclosed region. Applications to area and flux.
L17
Stokes' Theorem and the Divergence Theorem
Stokes as the 3D generalization of Green's. The Divergence Theorem relating a surface integral to a volume integral. The unified view: all four big theorems as instances of one generalization.
L18 (cap)
Capstone: Exam Simulation
A full-length Calc III simulation. Every missed problem corrected with reasoning.
Assignment 6
A full-length simulated Calc III final covering all 6 units. Every missed problem gets a written correction with the reasoning behind the right approach.
You complete Calculus III with a complete command of multivariable and vector calculus.
02Reference
Core Concepts
•
Dot Product
a·b = |a||b|cosθ. The dot product is zero when vectors are perpendicular. It measures how much one vector projects onto another. It is a scalar, not a vector.
Cross product · Projection · Work
∇f
The Gradient
The gradient of f is a vector pointing in the direction of steepest ascent. Its magnitude is the rate of increase in that direction. The gradient is always perpendicular to level curves.
Directional derivative · Partial derivatives · Chain rule
∂f/∂x
Partial Derivatives
The partial derivative with respect to x treats all other variables as constants. It measures the rate of change of f in the x-direction only. Clairaut's theorem: mixed partials are equal under mild conditions.
Gradient · Chain rule · Implicit differentiation
∬
Double Integrals
A double integral integrates over a 2D region. Iterated integrals compute it by integrating one variable at a time. The order of integration can sometimes be reversed to make the integral tractable.
Polar coordinates · Jacobian · Change of variables
∮
Green's Theorem
Green's theorem relates a line integral around a closed curve to a double integral over the enclosed region. It is the 2D version of the Fundamental Theorem of Calculus.
Stokes' theorem · Divergence theorem · Conservative fields
∇×F
Curl and Divergence
Curl measures the rotation of a vector field at a point. Divergence measures the expansion or contraction. A field is conservative if and only if its curl is zero.
Stokes' theorem · Flux · Conservative fields
03Capstone
Simulated Final Exam

The capstone covers all six units of multivariable and vector calculus. Every missed problem gets a written correction explaining the geometric reasoning that should have driven the approach.

○
Gap Review
Two weakest units revisited before the simulation.
○
Full Exam Simulation
All 6 units, timed, independent.
○
Written Corrections
Every miss corrected with geometric reasoning.
○
Diff Eq Readiness
A final session on what differential equations will require.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
Popular
per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x/week draws down lessons faster — no extra charge.

Get started
Semester
$3,775
26 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc
  • Session notes after every lesson
  • Capstone review + corrections
  • Next track readiness session

The full arc in one commitment.

Enroll now
In-person sessions available on Monthly and Semester plans at $800/mo and $4,625/semester. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Starting Calc III or need a structured second pass? One session is enough to figure out where you are and what the path looks like.

What do I need coming in? +
Solid Calc II: integration techniques, sequences and series, and comfort with the definite integral as a tool. Polar coordinates will help.
Is this the same as Multivariable Calculus? +
Yes. Calc III, Multivariable Calculus, and Calculus of Several Variables are different names for the same course at different schools.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.