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Calculus

| nosplit | ↑ parent "Mathematics" | words: 17 | descendant words: 9k | descendants: 216

nosplit | ⇓ toc | ↑ parent "Mathematics" | Wikipedia | words: 17 | descendant words: 9k | descendants: 216

Well summarized as "the branch of mathematics that deals with limits.

Table of contents | 17, 9k, 216
- Mathematical analysis | 17 | 🔗 link | nosplit | ↑ parent "Calculus"
- Limit (mathematics) | 46, 250, 7 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Convergent series | 🔗 link | nosplit | ↑ parent "Limit"
  - Continuous function | 0, 116, 3 | 🔗 link | nosplit | ↑ parent "Limit"
    - Continuous problems are simpler than discrete ones | 92 | 🔗 link | nosplit | ↑ parent "Continuous function"
    - Discrete | 24, 24, 1 | 🔗 link | nosplit | ↑ parent "Continuous function"
      - Discretization | 🔗 link | nosplit | ↑ parent "Discrete"
  - Infinity ( $\infty$ ) | 88 | 🔗 link | nosplit | ↑ parent "Limit"
  - L'Hôpital's rule (limit of a ratio) | 🔗 link | nosplit | ↑ parent "Limit"
- Derivative | 0, 266, 18 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Chain rule | 0, 0, 1 | 🔗 link | nosplit | ↑ parent "Derivative"
    - Multivariable chain rule | 🔗 link | nosplit | ↑ parent "Chain rule"
  - Differentiable function | 0, 4, 4 | 🔗 link | nosplit | ↑ parent "Derivative"
    - Smoothness | 🔗 link | nosplit | ↑ parent "Differentiable function"
    - Infinitely differentiable function ( $C^{\infty}$ ) | 0, 4, 2 | 🔗 link | nosplit | ↑ parent "Differentiable function"
      - Bump function | 0, 4, 1 | 🔗 link | nosplit | ↑ parent "Infinitely differentiable function"
        
        Flat top bump function | 4 | 🔗 link | nosplit | ↑ parent "Bump function"
  - Maxima and minima | 0, 4, 2 | 🔗 link | nosplit | ↑ parent "Derivative"
    - Lifegard problem | 4 | 🔗 link | nosplit | ↑ parent "Maxima and minima"
    - Saddle point | 🔗 link | nosplit | ↑ parent "Maxima and minima"
  - Newton dot notation | 🔗 link | nosplit | ↑ parent "Derivative"
  - Partial derivative | 0, 188, 4 | 🔗 link | nosplit | ↑ parent "Derivative"
    - Partial derivative notation | 0, 188, 3 | 🔗 link | nosplit | ↑ parent "Partial derivative"
  - Total derivative | 70 | 🔗 link | nosplit | ↑ parent "Derivative"
  - Directional derivative | 🔗 link | nosplit | ↑ parent "Derivative"
- Integral | 0, 1k, 19 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Area | 0, 15, 1 | 🔗 link | nosplit | ↑ parent "Integral"
    - Volume | 15 | 🔗 link | nosplit | ↑ parent "Area"
  - Riemann integral | 26 | 🔗 link | nosplit | ↑ parent "Integral"
  - Lebesgue integral | 91, 1k, 15 | 🔗 link | nosplit | ↑ parent "Integral"
    - Lebesgue integral vs Riemann integral | 201, 272, 1 | 🔗 link | nosplit | ↑ parent "Lebesgue integral"
      - Real world applications of the Lebesgue integral | 71 | 🔗 link | nosplit | ↑ parent "Lebesgue integral vs Riemann integral"
    - Lebesgue measurable | 🔗 link | nosplit | ↑ parent "Lebesgue integral"
    - Lebesgue integral of $L^{p}$ is complete but Riemann isn't | 91, 805, 11 | 🔗 link | nosplit | ↑ parent "Lebesgue integral"
      - Riesz-Fischer theorem | 59, 306, 4 | 🔗 link | nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        $L^{p}$ is complete | 1 | 🔗 link | nosplit | ↑ parent "Riesz-Fischer theorem"
        
        Fourier basis is complete for $L^{2}$ | 62, 246, 2 | 🔗 link | nosplit | ↑ parent "Riesz-Fischer theorem"
        
        $L^{p}$ norm sequence convergence does not imply pointwise convergence | 44 | 🔗 link | nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
        
        Carleson's theorem | 140 | 🔗 link | nosplit | ↑ parent "Fourier basis is complete for $L^{2}$ "
      - Lp space ( $L^{p}$ ) | 30, 408, 5 | 🔗 link | nosplit | ↑ parent "Lebesgue integral of $L^{p}$ is complete but Riemann isn't"
        
        $L^{1}$ | 🔗 link | nosplit | ↑ parent "Lp space"
        
        $L^{2}$ | 130, 378, 3 | 🔗 link | nosplit | ↑ parent "Lp space"
        
        Plancherel theorem | 150, 248, 2 | 🔗 link | nosplit | ↑ parent " $L^{2}$ "
        
        The Fourier transform is a bijection in $L^{2}$ | 87 | 🔗 link | nosplit | ↑ parent "Plancherel theorem"
        
        Every Riemann integrable function is Lebesgue integrable | 11 | 🔗 link | nosplit | ↑ parent "Plancherel theorem"
- Measure theory | 57 | 🔗 link | nosplit | ↑ parent "Calculus"
- Fourier series | 143, 555, 8 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Applications of the Fourier series | 0, 74, 1 | 🔗 link | nosplit | ↑ parent "Fourier series"
    - Solving partial differential equations with the Fourier series | 74 | 🔗 link | nosplit | ↑ parent "Applications of the Fourier series"
  - Discrete Fourier transform (DFT) | 🔗 link | nosplit | ↑ parent "Fourier series"
  - Fourier transform | 186, 321, 3 | 🔗 link | nosplit | ↑ parent "Fourier series"
    - Multidimensional Fourier transform | 43 | 🔗 link | nosplit | ↑ parent "Fourier transform"
    - Fourier inversion theorem | 41 | 🔗 link | nosplit | ↑ parent "Fourier transform"
    - Laplace transform | 51 | 🔗 link | nosplit | ↑ parent "Fourier transform"
  - History of the Fourier series | 17 | 🔗 link | nosplit | ↑ parent "Fourier series"
- Topology | 152, 2k, 61 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Covering space | 84, 84, 1 | 🔗 link | nosplit | ↑ parent "Topology"
    - Double cover | 🔗 link | nosplit | ↑ parent "Covering space"
  - Neighbourhood (mathematics) | 12 | 🔗 link | nosplit | ↑ parent "Topology"
  - Topological space | 🔗 link | nosplit | ↑ parent "Topology"
  - Manifold | 164, 424, 6 | 🔗 link | nosplit | ↑ parent "Topology"
    - Atlas (topology) | 25, 25, 1 | 🔗 link | nosplit | ↑ parent "Manifold"
      - Coordinate chart | 🔗 link | nosplit | ↑ parent "Atlas"
    - Covariant derivative | 39 | 🔗 link | nosplit | ↑ parent "Manifold"
    - Tangent space | 136, 148, 1 | 🔗 link | nosplit | ↑ parent "Manifold"
      - Tangent vector to a manifold | 12 | 🔗 link | nosplit | ↑ parent "Tangent space"
    - One-form | 48 | 🔗 link | nosplit | ↑ parent "Manifold"
  - Metric (mathematics, $d (x, y)$ ) | 106, 472, 10 | 🔗 link | nosplit | ↑ parent "Topology"
    - Metric space | 10, 366, 9 | 🔗 link | nosplit | ↑ parent "Metric"
      - Metric space vs normed vector space vs inner product space | 92 | 🔗 link | nosplit | ↑ parent "Metric space"
      - Complete metric space | 44 | 🔗 link | nosplit | ↑ parent "Metric space"
      - Normed vector space | 0, 49, 2 | 🔗 link | nosplit | ↑ parent "Metric space"
        
        Inner product space | 23, 49, 1 | 🔗 link | nosplit | ↑ parent "Normed vector space"
        
        Inner product | 26 | 🔗 link | nosplit | ↑ parent "Inner product space"
      - Norm (mathematics, $∣ x ∣$ ) | 78, 145, 2 | 🔗 link | nosplit | ↑ parent "Metric space"
        
        Norm induced by an inner product | 26 | 🔗 link | nosplit | ↑ parent "Norm"
        
        Metric induced by a norm | 41 | 🔗 link | nosplit | ↑ parent "Norm"
      - Pseudometric space | 26 | 🔗 link | nosplit | ↑ parent "Metric space"
  - Compact space | 🔗 link | nosplit | ↑ parent "Topology"
  - Dense set | 🔗 link | nosplit | ↑ parent "Topology"
  - Connected space | 0, 39, 3 | 🔗 link | nosplit | ↑ parent "Topology"
    - Connected component | 39 | 🔗 link | nosplit | ↑ parent "Connected space"
    - Simply connected space | 0, 0, 1 | 🔗 link | nosplit | ↑ parent "Connected space"
      - Loop (topology) | 🔗 link | nosplit | ↑ parent "Simply connected space"
  - Homotopy | 0, 526, 7 | 🔗 link | nosplit | ↑ parent "Topology"
    - Generalized Poincaré conjecture | 282, 526, 6 | 🔗 link | nosplit | ↑ parent "Homotopy"
      - Exotic sphere | 🔗 link | nosplit | ↑ parent "Generalized Poincaré conjecture"
      - Poincaré conjecture | 🔗 link | nosplit | ↑ parent "Generalized Poincaré conjecture"
      - Classification of closed surfaces | 215, 244, 3 | 🔗 link | nosplit | ↑ parent "Generalized Poincaré conjecture"
        
        Torus | 🔗 link | nosplit | ↑ parent "Classification of closed surfaces"
        
        Möbius strip | 🔗 link | nosplit | ↑ parent "Classification of closed surfaces"
        
        Klein bottle | 29 | 🔗 link | nosplit | ↑ parent "Classification of closed surfaces"
  - Real coordinate space ( $R^{n}$ ) | 0, 213, 24 | 🔗 link | nosplit | ↑ parent "Topology"
    - Real line ( $R^{1}$ ) | 🔗 link | nosplit | ↑ parent "Real coordinate space"
    - Real plane ( $R^{2}$ ) | 🔗 link | nosplit | ↑ parent "Real coordinate space"
    - Real coordinate space of dimension three ( $R^{3}$ ) | 🔗 link | nosplit | ↑ parent "Real coordinate space"
    - Real coordinate space of dimension four ( $R^{4}$ , 4D) | 10, 28, 1 | 🔗 link | nosplit | ↑ parent "Real coordinate space"
      - Visualizing 4D | 18 | 🔗 link | nosplit | ↑ parent "Real coordinate space of dimension four"
    - Dimension | 0, 4, 2 | 🔗 link | nosplit | ↑ parent "Real coordinate space"
      - Infinite dimensional | 4, 4, 1 | 🔗 link | nosplit | ↑ parent "Dimension"
        
        Finite dimensional | 🔗 link | nosplit | ↑ parent "Infinite dimensional"
    - Complex coordinate space ( $C^{n}$ ) | 0, 115, 1 | 🔗 link | nosplit | ↑ parent "Real coordinate space"
      - Complex dot product | 115 | 🔗 link | nosplit | ↑ parent "Complex coordinate space"
    - Euclidean space | 23, 66, 13 | 🔗 link | nosplit | ↑ parent "Real coordinate space"
      - Euclidean metric signature matrix | 9 | 🔗 link | nosplit | ↑ parent "Euclidean space"
      - Cartesian coordinate system | 🔗 link | nosplit | ↑ parent "Euclidean space"
      - Polar coordinate system | 0, 0, 1 | 🔗 link | nosplit | ↑ parent "Euclidean space"
        
        Spherical coordinate system | 🔗 link | nosplit | ↑ parent "Polar coordinate system"
      - Pythagorean theorem | 🔗 link | nosplit | ↑ parent "Euclidean space"
      - Non-Euclidean geometry | 0, 34, 7 | 🔗 link | nosplit | ↑ parent "Euclidean space"
        
        Elliptic geometry | 0, 34, 2 | 🔗 link | nosplit | ↑ parent "Non-Euclidean geometry"
        
        Model of elliptic geometry | 0, 34, 1 | 🔗 link | nosplit | ↑ parent "Elliptic geometry"
        
        Projective elliptic geometry | 34 | 🔗 link | nosplit | ↑ parent "Model of elliptic geometry"
        
        Hyperbolic gemoetry | 0, 0, 3 | 🔗 link | nosplit | ↑ parent "Non-Euclidean geometry"
        
        Hyperbolic functions | 0, 0, 2 | 🔗 link | nosplit | ↑ parent "Hyperbolic gemoetry"
        
        Hyperbolic sine | 🔗 link | nosplit | ↑ parent "Hyperbolic functions"
        
        Hyperbolic cossine | 🔗 link | nosplit | ↑ parent "Hyperbolic functions"
- Distribution (mathematics) | 182, 265, 3 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Dirac delta function | 83, 83, 2 | 🔗 link | nosplit | ↑ parent "Distribution"
    - Green's function | 🔗 link | nosplit | ↑ parent "Dirac delta function"
    - Heaviside step function | 🔗 link | nosplit | ↑ parent "Dirac delta function"
- Complex analysis | 14, 277, 6 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Holomorphic function | 69 | 🔗 link | nosplit | ↑ parent "Complex analysis"
  - Analytic continuation | 52, 194, 4 | 🔗 link | nosplit | ↑ parent "Complex analysis"
    - Visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) | 16 | 🔗 link | nosplit | ↑ parent "Analytic continuation"
    - Identity theorem | 92, 126, 2 | 🔗 link | nosplit | ↑ parent "Analytic continuation"
      - Riemann zeta function | 0, 34, 1 | 🔗 link | nosplit | ↑ parent "Identity theorem"
        
        Riemann hypothesis | 34 | 🔗 link | nosplit | ↑ parent "Riemann zeta function"
- Hilbert space | 34, 184, 1 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Complete basis | 150 | 🔗 link | nosplit | ↑ parent "Hilbert space"
- Differential equation | 0, 3k, 69 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Euler number ( $e$ ) | 0, 11, 2 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - Natural logarithm ( $l n (n)$ , $l o g_{e} (n)$ ) | 0, 11, 1 | 🔗 link | nosplit | ↑ parent "Euler number"
      - Euler-Mascheroni constant | 11 | 🔗 link | nosplit | ↑ parent "Natural logarithm"
  - Linear differential equation | 172, 172, 1 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - Holonomic function | 🔗 link | nosplit | ↑ parent "Linear differential equation"
  - Ordinary differential equation (ODE) | 0, 0, 2 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - System of ordinary differential equations | 0, 0, 1 | 🔗 link | nosplit | ↑ parent "Ordinary differential equation"
      - System of linear ordinary differential equations | 🔗 link | nosplit | ↑ parent "System of ordinary differential equations"
  - Partial differential equation (PDE) | 0, 2k, 45 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - Analytical method to solve a partial differential equation | 6, 88, 1 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
      - Separation of variables | 82 | 🔗 link | nosplit | ↑ parent "Analytical method to solve a partial differential equation"
    - Numerical method to solve a partial differential equation | 20, 79, 3 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
      - Variational formulation of a partial differential equation | 32, 32, 1 | 🔗 link | nosplit | ↑ parent "Numerical method to solve a partial differential equation"
        
        Weak solution | 🔗 link | nosplit | ↑ parent "Variational formulation of a partial differential equation"
      - Finite element method | 27 | 🔗 link | nosplit | ↑ parent "Numerical method to solve a partial differential equation"
    - Important partial differential equation | 43, 793, 24 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
      - Laplace's equation | 83, 188, 5 | 🔗 link | nosplit | ↑ parent "Important partial differential equation"
        
        Legendre polynomials | 46 | 🔗 link | nosplit | ↑ parent "Laplace's equation"
        
        Poisson's equation | 16, 16, 1 | 🔗 link | nosplit | ↑ parent "Laplace's equation"
        
        Uniqueness theorem for Poisson's equation | 🔗 link | nosplit | ↑ parent "Poisson's equation"
        
        Harmonic function | 11, 43, 1 | 🔗 link | nosplit | ↑ parent "Laplace's equation"
        
        Spherical harmonic | 32 | 🔗 link | nosplit | ↑ parent "Harmonic function"
      - Heat equation | 144 | 🔗 link | nosplit | ↑ parent "Important partial differential equation"
      - Wave equation | 124, 418, 16 | 🔗 link | nosplit | ↑ parent "Important partial differential equation"
        
        The wave equation can be seen as infinitely many infinitesimal coupled oscillators | 67 | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Wave | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Diffraction | 0, 57, 5 | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Arago spot | 🔗 link | nosplit | ↑ parent "Diffraction"
        
        Huygens-Fresnel principle | 0, 57, 3 | 🔗 link | nosplit | ↑ parent "Diffraction"
        
        Kirchhoff's diffraction formula | 10, 57, 2 | 🔗 link | nosplit | ↑ parent "Huygens-Fresnel principle"
        
        Fraunhofer diffraction | 24 | 🔗 link | nosplit | ↑ parent "Kirchhoff's diffraction formula"
        
        Fresnel diffraction | 23 | 🔗 link | nosplit | ↑ parent "Kirchhoff's diffraction formula"
        
        Refraction | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Resonance | 8 | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Wave interference | 0, 33, 1 | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Interference pattern | 33 | 🔗 link | nosplit | ↑ parent "Wave interference"
        
        2D wave equation on a circular domain | 0, 112, 2 | 🔗 link | nosplit | ↑ parent "Wave equation"
        
        Bessel function | 86, 112, 1 | 🔗 link | nosplit | ↑ parent "2D wave equation on a circular domain"
        
        Fourier-Bessel series | 26 | 🔗 link | nosplit | ↑ parent "Bessel function"
        
        Helmholtz equation | 17 | 🔗 link | nosplit | ↑ parent "Wave equation"
    - Existence and uniqueness of solutions of partial differential equations | 116 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
    - Partial differential equation solver | 0, 1k, 6 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
      - FreeFem | 642, 685, 3 | 🔗 link | nosplit | ↑ parent "Partial differential equation solver"
        
        FreeFem examples | 0, 43, 2 | 🔗 link | nosplit | ↑ parent "FreeFem"
        
        heat-dirichlet.1d.freefem | 21 | 🔗 link | nosplit | ↑ parent "FreeFem examples"
        
        heat-dirichlet-2d-freefem | 22 | 🔗 link | nosplit | ↑ parent "FreeFem examples"
      - FEniCS Project | 233, 372, 1 | 🔗 link | nosplit | ↑ parent "Partial differential equation solver"
        
        Hans Petter Langtangen | 139 | 🔗 link | nosplit | ↑ parent "FEniCS Project"
    - System of partial differential equations | 77 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
    - Classification of second order partial differential equations into elliptic, parabolic and hyperbolic | 43, 123, 4 | 🔗 link | nosplit | ↑ parent "Partial differential equation"
  - Phase space | 23 | 🔗 link | nosplit | ↑ parent "Differential equation"
  - Boundary condition | 0, 469, 9 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - Initial condition | 29 | 🔗 link | nosplit | ↑ parent "Boundary condition"
    - Boundary value problem | 🔗 link | nosplit | ↑ parent "Boundary condition"
    - Dirichlet boundary condition | 46 | 🔗 link | nosplit | ↑ parent "Boundary condition"
    - Neumann boundary condition | 28, 343, 4 | 🔗 link | nosplit | ↑ parent "Boundary condition"
      - Cauchy boundary condition | 70 | 🔗 link | nosplit | ↑ parent "Neumann boundary condition"
      - Robin boundary condition | 117 | 🔗 link | nosplit | ↑ parent "Neumann boundary condition"
      - Open boundary condition | 115 | 🔗 link | nosplit | ↑ parent "Neumann boundary condition"
      - Mixed boundary condition | 13 | 🔗 link | nosplit | ↑ parent "Neumann boundary condition"
    - Time dependent boundary condition | 51 | 🔗 link | nosplit | ↑ parent "Boundary condition"
  - Control theory | 104, 104, 3 | 🔗 link | nosplit | ↑ parent "Differential equation"
    - Control engineering | 🔗 link | nosplit | ↑ parent "Control theory"
    - Control system | 🔗 link | nosplit | ↑ parent "Control theory"
    - Feedback loop | 🔗 link | nosplit | ↑ parent "Control theory"
- Series (mathematics) | 0, 0, 3 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Power series | 0, 0, 2 | 🔗 link | nosplit | ↑ parent "Series"
    - Radius of convergence | 🔗 link | nosplit | ↑ parent "Power series"
    - Taylor series | 🔗 link | nosplit | ↑ parent "Power series"
- Gradient, Divergence, Curl, and Laplacian | 0, 401, 7 | 🔗 link | nosplit | ↑ parent "Calculus"
  - Curl (mathematics, $\nabla \times$ ) | 11 | 🔗 link | nosplit | ↑ parent "Gradient, Divergence, Curl, and Laplacian"
  - Nabla symbol ( $\nabla$ ) | 96, 149, 1 | 🔗 link | nosplit | ↑ parent "Gradient, Divergence, Curl, and Laplacian"
    - Del | 53 | 🔗 link | nosplit | ↑ parent "Nabla symbol"
  - Divergence ( $\nabla \cdot$ , $d i v ()$ ) | 74 | 🔗 link | nosplit | ↑ parent "Gradient, Divergence, Curl, and Laplacian"
  - Gradient ( $\nabla$ ) | 89 | 🔗 link | nosplit | ↑ parent "Gradient, Divergence, Curl, and Laplacian"
  - Laplace operator ( $Δ$ , $\nabla^{2}$ ) | 39, 78, 1 | 🔗 link | nosplit | ↑ parent "Gradient, Divergence, Curl, and Laplacian"
    - D'alembert operator ( $□$ ) | 39 | 🔗 link | nosplit | ↑ parent "Laplace operator"
- Infinitesimal | 23 | 🔗 link | nosplit | ↑ parent "Calculus"

Ancestors

Mathematics | 17, 34k, 763
Ciro Santilli's Homepage | 262, 237k, 4k

Incoming links

Limit | 46, 250, 7
Mathematical analysis | 17
SymPy | 626
Topology | 152, 2k, 61