MathsTricks & Calculus

Advanced Calculus Multivariable
Calculus

Calculus doesn't stop at one dimension. When functions depend on two or three variables, derivatives become gradients, integrals span regions and volumes, and the Fundamental Theorem generalises into three beautiful theorems about fields and surfaces.

10In-depth Topics
100+Worked Examples
100Quiz Questions
3Great Theorems
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Home › Multivariable Calculus
What you'll study

From f(x) to
f(x, y, z)

Single-variable calculus asks how a function changes as one input varies. Multivariable calculus asks the same question when the function depends on two, three, or more inputs simultaneously, which is the reality of almost every physical, economic, and engineering model.

You'll extend every tool you know: derivatives become partial derivatives and gradient vectors, integrals span regions, volumes, curves, and surfaces, and the Fundamental Theorem of Calculus becomes three distinct theorems (Green's, Stokes', and the Divergence Theorem) each linking an integral over a region to an integral over its boundary.

Prerequisites: Chain Rule Integration Definite Integrals Vectors
By the end of this section, you can:
  • Compute partial derivatives and interpret them geometrically as slopes in coordinate directions
  • Find the gradient ∇f and use it to identify the direction of steepest ascent
  • Apply the multivariable chain rule using dependency tree diagrams
  • Set up and evaluate double and triple integrals over general regions
  • Convert integrals to polar, cylindrical and spherical coordinates
  • Compute line integrals of scalar functions and work done by vector fields
  • Evaluate surface integrals and flux through oriented surfaces
  • Apply Green's, Stokes', and Divergence theorems to simplify calculations
The big picture

How single-variable ideas generalise

Every concept you know from calculus has a multivariable counterpart. The leap is smaller than it looks.

Single variable
Derivative f′(x)
Rate of change along the single input axis.
dy/dx
→
Multivariable
Partial Derivatives
Rate of change in each coordinate direction, holding others constant.
∂f/∂x,   ∂f/∂y
→
All at once
The Gradient ∇f
A vector encoding every partial derivative — points in the direction of steepest ascent.
∇f = (∂f/∂x, ∂f/∂y)
Single variable
Definite Integral
Area under a curve over an interval.
∫ₐᵇ f(x) dx
→
Two variables
Double Integral
Volume under a surface over a 2D region.
∬R f(x,y) dA
→
Three variables
Triple Integral
Mass, charge, or any accumulated quantity over a 3D volume.
∭V f dV
Single variable
FTC
∫ₐᵇ f′dx = f(b)−f(a). Integral over region = boundary values.
∫ₐᵇ f′ dx = f(b)−f(a)
→
2D version
Green's Theorem
Double integral over region = line integral around boundary.
∬D(∂Q/∂x−∂P/∂y)dA = ∮ P dx+Q dy
→
3D versions
Stokes' & Divergence
Surface integral / volume integral = boundary integral.
∭(∇·F)dV = ∯ F·dS
All topics

10 In-Depth Topics

Each page has 1,000–2,000 words of explanation, 10 or more fully worked examples, and a 10-question interactive quiz.

Topic 01 — Start Here
Partial Derivatives
The foundation of everything. Learn to differentiate a function of two or more variables with respect to each variable independently, how to compute all four second-order partials, and why Clairaut's theorem guarantees fxy = fyx for smooth functions.
∂f/∂x Higher-order partials Clairaut's theorem Geometric slopes
12 examples 10-Q quiz ~1,500 words
→
Topic 02
Gradient & Directional Derivatives
The gradient ∇f is the vector of all partial derivatives. It points in the direction of steepest ascent. Learn the directional derivative formula Duf = ∇f · û and the deep connection between ∇f and level curves.
∇f gradient Directional derivative Level curves Max rate of change
10 examples10-Q quiz
→
Topic 03
Chain Rule for Multivariable Functions
When z = f(x,y) and x,y both depend on t, the total derivative dz/dt is found by summing partial derivatives along each path of the dependency tree. Extends naturally to implicit differentiation via partials.
Dependency tree Total derivative dz/dt Implicit diff via ∂
10 examples10-Q quiz
→
Topic 04
Double Integrals
Extend the definite integral to functions of two variables. Master Fubini's theorem for iterated integrals, setting up limits for Type I and Type II regions, switching the order of integration, and converting to polar coordinates for circular domains.
Fubini's theorem Type I & II regions Order of integration Polar form
12 examples10-Q quiz
→
Topic 05
Polar Coordinates
When a region or curve has circular symmetry, polar coordinates r and θ simplify everything. Learn conversions, common polar curves (cardioid, rose, lemniscate), area via ½∫r² dθ, and arc length in polar form.
r = f(θ) Cardioid & rose curves Polar area Arc length
10 examples10-Q quiz
→
Topic 06
Triple Integrals
Compute volume, mass, and charge density over three-dimensional regions. Learn to set up iterated triple integrals in Cartesian coordinates, convert to cylindrical coordinates (r, θ, z) for cylindrical domains, and to spherical coordinates (ρ, θ, φ) for spherical domains.
∭ f dV Cylindrical coords Spherical coords Jacobians
10 examples10-Q quiz
→
Topic 07
Cylindrical & Spherical Coordinates
A deep dive into 3D coordinate systems built for symmetry. Cylindrical (r,θ,z) for rotation around the z-axis; spherical (ρ,θ,φ) for balls and shells. Derivation of the Jacobians r and ρ²sin φ and when to use each system.
Cylindrical (r,θ,z) Spherical (ρ,θ,φ) Jacobian ρ²sinφ
10 examples10-Q quiz
→
Topic 08
Line Integrals
Integrate along a curve in space rather than over an interval. Scalar line integrals ∫C f ds give arc-length-weighted totals; vector line integrals ∫C F·dr compute the work done by a force field along a path. Includes path parameterisation and the concept of conservative fields.
∫_C f ds Work integral F·dr Parameterisation Conservative fields
10 examples10-Q quiz
→
Topic 09
Surface Integrals
Integrate over a 2D surface embedded in 3D space. Scalar surface integrals measure total mass or area; vector surface integrals (flux integrals) ∬S F·dS measure how much of a field passes through an oriented surface — the physical basis of Gauss's law and fluid flow.
Surface parameterisation dS surface element Flux ∬ F·dS
10 examples10-Q quiz
→
Topic 10 — Grand Finale
Green's, Stokes' & Divergence Theorems
The three great unifying theorems of vector calculus, all generalisations of the Fundamental Theorem: Green's links a double integral over a region to a line integral around its boundary; Stokes' links a surface integral of curl to a boundary line integral; the Divergence Theorem links a volume integral of divergence to a surface integral of flux. Together they form one of the most beautiful structures in mathematics.
Green's theorem Stokes' theorem Divergence theorem ∇× curl ∇· divergence
10 examples10-Q quiz~2,000 words
→
Recommended order

Suggested Learning Pathway

Topics build on one another. Follow this order for the smoothest experience, though each page is designed to stand alone.

∂
Partial
Derivatives
∇
Gradient &
Directional
dz/dt
Multivariable
Chain Rule
r,θ
Polar
Coords
∬
Double
Integrals
ρ,φ
3D
Coordinates
∭
Triple
Integrals
∫C
Line
Integrals
∬S
Surface
Integrals
∮
The Great
Theorems
Essential formulas

Key Results at a Glance

The most important formulas in multivariable calculus, each one the headline of a full topic page.

Partial Derivatives
Definition of ∂f/∂x
∂f/∂x = limh→0 [f(x+h,y) − f(x,y)] / h
Differentiate with respect to x, treating y as a constant.
Gradient & Directional Derivative
Directional Derivative
Dûf = ∇f · û = |∇f| cos θ
Maximum when û points in the direction of ∇f; zero along level curves.
Chain Rule
Total Derivative
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
Sum over all paths in the dependency tree.
Double Integrals
Fubini's Theorem
∬R f dA = ∫∫ f(x,y) dy dx
Integrate in either order when f is continuous on R.
Polar Coordinates
Area Element
dA = r dr dθ  (Jacobian = r)
The extra r factor comes from the area of a polar "wedge".
Spherical Coordinates
Volume Element
dV = ρ² sin φ dρ dφ dθ
The Jacobian ρ²sin φ from the coordinate transformation.
Line Integrals
Work by a Vector Field
W = ∫C F · dr = ∫ab F(r(t)) · r′(t) dt
Dot product of field with tangent vector, integrated over the path.
Green's Theorem
2D Fundamental Theorem
∮∂D P dx + Q dy = ∬D(∂Q/∂x − ∂P/∂y) dA
Boundary line integral equals interior double integral of curl.
Divergence Theorem
Gauss's Theorem
∯∂V F · dS = ∭V (∇ · F) dV
Total flux out of a closed surface equals the divergence integrated over the enclosed volume.
Common questions

Before You Start

Answers to the questions students ask most often.

Do I need single-variable calculus first?
Yes. You should be comfortable with derivatives (chain rule, implicit differentiation), definite integrals, and substitution. A good working knowledge of vectors is also expected from Topic 07 onwards.
What's the difference between a partial derivative and a total derivative?
A partial derivative ∂f/∂x holds all other variables constant. The total derivative (or directional derivative) accounts for simultaneous change in all variables, encoded compactly by the gradient vector ∇f.
Why do we need different coordinate systems?
Cartesian coordinates (x,y,z) work for rectangular problems. Polar and cylindrical coordinates exploit rotational symmetry; spherical coordinates suit spheres and cones. Choosing the right system transforms a painful integral into a clean one.
What is a Jacobian and when do I use it?
The Jacobian is a scaling factor that corrects the area or volume element when you change coordinate systems. For polar coordinates, dA = r dr dθ — the r is the Jacobian. For spherical, dV = ρ²sin φ dρ dφ dθ.
Are Green's, Stokes', and Divergence theorems really all the same theorem?
Essentially yes. All three are special cases of the general Stokes' theorem from differential geometry: ∫∂M ω = ∫M dω. They all say: an integral over a region equals an integral over its boundary, with differentiation (curl or divergence) bridging the two.
What can I do with multivariable calculus in practice?
It underpins electromagnetism (Maxwell's equations use divergence and curl), fluid mechanics (Navier–Stokes equations), heat transfer, computer graphics (surface normals, ray casting), machine learning (gradient descent), economics (constrained optimisation), and much more.

Ready to Begin?

Start with partial derivatives (the single most important concept in the entire section) and work your way through to the grand theorems.

Start: Partial Derivatives → Jump to the Theorems

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