maths.free › Multivariable Calculus

Multivariable Calculus

Calculus in more than one dimension: surfaces instead of curves, gradients instead of slopes, volumes instead of areas. Everything here is drawn in 3D — a surface you can rotate is worth a page of symbols.

@ action

Introductory Functions of several variables Surfaces, level curves, domains — reading z = f(x, y). x^2 - y^2 Core Partial derivatives and the gradient Differentiating in one direction at a time; the gradient as the vector of slopes. gradient of x^2*y - 3x + y^3 Core Optimisation in several variables Critical points, the second-derivative test, and Lagrange multipliers. gradient of x^2 + y^2 - 2x - 4y Core Double and triple integrals Volume under a surface by iterated integration; changing the order and the coordinates. integrate x^2 dx from 0 to 1 Core Vector fields, line integrals and the big theorems Gradient fields, curl and divergence, Green, Stokes and Gauss. gradient of x^2 + y^2

Chapters from Boelkins et al., Active Calculus Multivariable

Every section of the book, condensed into a lesson with its own practice problems.

1. Precalculus of Multivariable Functions

Why Multivariable?Three Dimensional SpaceVectorsLines in SpacePlanes in SpaceCommon Graphs in Three DimensionsPolar, Cylindrical, and Spherical Coordinates

2. Vector-Valued Functions of One Variable

Classic Calculus ApproachVector-Valued Functions of One VariableCalculus of Vector-Valued Functions of One VariableArc LengthThe TNB FrameCurvatureSplitting the Acceleration Vector

3. Derivatives of Multivariable Functions

Calculus of Several VariablesLimitsFirst-Order Partial DerivativesSecond-Order Partial DerivativesLinearization: Tangent Planes and DifferentialsThe Multivariable Chain RuleHigher DimensionsOptimizationConstrained Optimization: Lagrange Multipliers

4. Multiple Integrals

Integrating Functions of Several VariablesDouble Riemann Sums and Double Integrals over RectanglesIterated IntegralsApplications of Double IntegralsChange of Variables

5. Vector Calculus

Integrating Multivariable Vector-Valued FunctionsThe Idea of a Line IntegralUsing Parameterizations to Calculate Line IntegralsPath-Independent Vector Fields and the Fundamental Theorem of Calculus for Line IntegralsLine Integrals of Scalar FunctionsThe Divergence of a Vector FieldThe Curl of a Vector FieldParameterizations of Surfaces and Surface AreaFlux IntegralsSurface Integrals of Scalar Valued Functions

Chapters from OpenStax Calculus Volume 3

Every section of the book, condensed into a lesson with its own practice problems.

1. Parametric Equations and Polar Coordinates

Parametric EquationsCalculus of Parametric CurvesPolar CoordinatesArea and Arc Length in Polar CoordinatesConic Sections

2. Vectors in Space

Vectors in the PlaneVectors in Three DimensionsThe Dot ProductThe Cross ProductEquations of Lines and Planes in SpaceQuadric SurfacesCylindrical and Spherical Coordinates

3. Vector-Valued Functions

Vector-Valued Functions and Space CurvesCalculus of Vector-Valued FunctionsArc Length and CurvatureMotion in Space

4. Differentiation of Functions of Several Variables

Limits and ContinuityTangent Planes and Linear ApproximationsThe Chain RuleDirectional Derivatives and the GradientMaxima/Minima ProblemsLagrange Multipliers

5. Multiple Integration

Double Integrals over Rectangular RegionsDouble Integrals over General RegionsDouble Integrals in Polar CoordinatesTriple IntegralsTriple Integrals in Cylindrical and Spherical CoordinatesCalculating Centers of Mass and Moments of InertiaChange of Variables in Multiple Integrals

6. Vector Calculus

Line IntegralsConservative Vector FieldsGreen’s TheoremDivergence and CurlSurface IntegralsStokes’ TheoremThe Divergence Theorem

Symbols used here

\frac{\partial f}{\partial x}
partial derivative
Derivative with respect to x, holding the other variables fixed.
\nabla f
gradient (nabla, del)
Vector of partial derivatives; points uphill.
\int f(x)\,dx,\ \int_a^b
integral
Antiderivative (indefinite) or signed area from a to b (definite).
\iint,\ \oint
double / contour integral
Integral over a region of the plane; integral around a closed curve.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.
\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|
dot product, norm
Σ u_i v_i; the length of v, √(v·v).

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen — the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

@ action

＋ Arithmetic𝑥 Algebra△ Geometry∿ Trigonometryƒ Precalculusσ Statistics & Probability⊢ Discrete Math & Logic∫ Calculus⊞ Linear Algebraℕ Number Theoryẏ Differential Equationsε Real Analysis⋔ Combinatorics & Graph Theory∈ Set Theory & Logic𝔾 Abstract Algebra𝑃 Probability Theory≈ Numerical Methodsℂ Complex Analysis◎ Topologyμ Measure Theory‖·‖ Functional Analysisκ Differential Geometryπ₁ Algebraic Topology→ Category Theory∞ Frontiers