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Calculus

Front matter

  • Introduction
  • Graph of content

Algebra

  • 1. Numbers
    • 1.1. Real numbers
    • 1.2. Cartesian form of complex numbers
    • 1.3. Polar form of complex numbers
    • 1.4. Linear systems in two variables
  • 2. Vector algebra
    • 2.1. Vectors
    • 2.2. Dot product
    • 2.3. Cross product
    • 2.4. Lines and planes

Single-variable functions

  • 3. Functions
    • 3.1. Functions of one variable
    • 3.2. Domain, range and graphs
    • 3.3. Limits and continuity
    • 3.4. Injectivity and surjectivity
    • 3.5. Nonlinear equations in one variable
  • 4. Differentiation
    • 4.1. Differentiability and derivatives
    • 4.2. Chain rule
    • 4.3. Product rule
    • 4.4. Extrema of functions
    • 4.5. Implicit differentiation
    • 4.6. Approximating functions
    • 4.7. Graphs of single-variable functions
  • 5. Integration
    • 5.1. Integrability and antiderivatives
    • 5.2. Substitution rule
    • 5.3. Integration by parts
    • 5.4. Improper integrals
    • 5.5. Approximating integrals

Sequences and series

  • 6. Sequences
    • 6.1. Sequences and their types
    • 6.2. Properties of sequences
    • 6.3. Convergence of sequences
  • 7. Series
    • 7.1. Series and their types
    • 7.2. Convergence of series
    • 7.3. Convergence tests
    • 7.4. Functions and series

Multivariable functions

  • 8. Concepts and structures
    • 8.1. Functions of two or more variables
    • 8.2. Domain, range and graphs
    • 8.3. Curves and surfaces
    • 8.4. Limits and continuity
    • 8.5. Level curves and sets
    • 8.6. Nonlinear systems in two variables
    • 8.7. Graphs of two-variable functions
    • 8.8. Coordinate systems
  • 9. Partial differentiation
    • 9.1. Differentiability and partial derivatives
    • 9.2. Directional derivative and gradient
    • 9.3. Locally extreme values
    • 9.4. Globally extreme values
    • 9.5. Constrained optimisation
    • 9.6. Approximating functions
  • 10. Multivariable integration
    • 10.1. Multiple and repeated integrals
    • 10.2. Coordinate transformations
    • 10.3. Surface integrals
    • 10.4. Line integrals
  • 11. Vector calculus
    • 11.1. Vector fields
    • 11.2. Curl, divergence and flux
    • 11.3. Green’s, divergence and Stokes’ theorem

Differential equations

  • 12. First-order ordinary differential equations
    • 12.1. Differential equations and initial value problems
    • 12.2. Direction fields and solutions
    • 12.3. Separable differential equations
    • 12.4. First-order linear differential equations
    • 12.5. Approximating solutions
  • 13. Second-order ordinary differential equations
    • 13.1. Homogeneous problems with constant coefficients
    • 13.2. Inhomogeneous problems with constant coefficients

Appendices

  • 14. Overviews from algebra
    • 14.1. Algebraic identities
    • 14.2. Limits
    • 14.3. Vector identities
  • 15. Overviews from single-variable functions
    • 15.1. Polynomials and power functions
    • 15.2. Exponential and logarithmic functions
    • 15.3. Trigonometric functions
    • 15.4. Other functions
  • 16. Overviews from sequences and series
    • 16.1. Special sequences
    • 16.2. Special series
  • 17. Overviews from multivariable functions
    • 17.1. Common curves
    • 17.2. Common surfaces
    • 17.3. Integral identities

Back matter

  • Index
  • Credits and License
  • Repository
  • Open issue

Index

Index

Symbols | A | C | D | E | F | I | M | N | O | P | R | S | U | V

Symbols

  • $i$-th component
  • $i$-th entry

A

  • algebraic multiplicity
  • angle

C

  • Cartesian equation, [1]
  • Cauchy-Schwarz inequality
  • column vector
  • complex conjugate
  • complex number
  • complex plane
  • components
  • cross product

D

  • determinant
  • directional vector, [1], [2]
  • distance
  • dot product

E

  • entries
  • Euler's formula

F

  • fundamental theorem of algebra

I

  • imaginary part
  • imaginary unit
  • inner product

M

  • multiplicity

N

  • norm
  • normal equation, [1]
  • normal vector, [1]

O

  • of a vector $\mathbf{w}$ onto a non-zero vector $\mathbf{v}$
  • onto
  • orthogonal
  • orthogonal projection

P

  • parametric vector equation, [1], [2]
  • polar form

R

  • real part

S

  • scalar multiple
  • scalars
  • size
  • sum

U

  • unit vector
  • unit vector in the direction of $\mathbf{v}$

V

  • vector, [1]

By Authors from Delft University of Technology, built with TeachBooks and Jupyter Book, CC BY 4.0

Last updated on March 27, 2026.