Expression
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Curve definition
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Variables
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Description
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Linear transformations
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![{\displaystyle L[y]=y^{(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55748cdb64e2e70170550b1601827405a7391b16) |
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Derivative ofnth order
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![{\displaystyle L[y]=\int _{a}^{t}y\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb616ec3149c278d4ed4d471906c2f3e485248a6) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Integral, area
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![{\displaystyle L[y]=y\circ f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adf8e1a8de11d2b9a58113939186edfd36eb5b78) |
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Composition operator
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![{\displaystyle L[y]={\frac {y\circ t+y\circ -t}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4053efbb3a2417b0187252ce8ce11943d1811df1) |
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Even component
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![{\displaystyle L[y]={\frac {y\circ t-y\circ -t}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aea9dbfed39f54e300c1b0ecb00212a12c0bb5d8) |
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Odd component
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![{\displaystyle L[y]=y\circ (t+1)-y\circ t=\Delta y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a59c37d0103cd9e45f3008be15be307a2121318b) |
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Difference operator
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![{\displaystyle L[y]=y\circ (t)-y\circ (t-1)=\nabla y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/285cc671ff18fcea194db0b07bef7943c5cc718f) |
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Backward difference (Nabla operator)
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![{\displaystyle L[y]=\sum y=\Delta ^{-1}y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2734b315faf5301c1b49a3d1920dd6b1bb4b378) |
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Indefinite sumoperator (inverse operator of difference)
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![{\displaystyle L[y]=-(py')'+qy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec732fb50af69f011ca9eca0fc1fc8156cfc954c) |
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Sturm–Liouville operator
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Non-linear transformations
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![{\displaystyle F[y]=y^{[-1]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e505c0ad5eebe33b1d91e82cada72c95c09d04c9) |
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Inverse function
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![{\displaystyle F[y]=t\,y'^{[-1]}-y\circ y'^{[-1]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3ef79233c2a76b9e7a6b2fe1ab9bd690af7e5a) |
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Legendre transformation
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![{\displaystyle F[y]=f\circ y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e5c7044beca35c385f28f9684ef56ccedcb4fc2) |
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Left composition
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![{\displaystyle F[y]=\prod y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6db0139615cdc804309a1e7360c50e6aeae0256e) |
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Indefinite product
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![{\displaystyle F[y]={\frac {y'}{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be39d46a4b51569e8fd4e481ed2935f4ae3e2197) |
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Logarithmic derivative
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![{\displaystyle F[y]={\frac {ty'}{y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33bad35cb3771f86c734f0123676152bae90fde6) |
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Elasticity
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![{\displaystyle F[y]={y''' \over y'}-{3 \over 2}\left({y'' \over y'}\right)^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82429a931a7be02e43c48a978c443a08f7563a1f) |
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Schwarzian derivative
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![{\displaystyle F[y]=\int _{a}^{t}|y'|\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2759c4bc022e78e98f4215212344e0abfa4f474d) |
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Total variation
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![{\displaystyle F[y]={\frac {1}{t-a}}\int _{a}^{t}y\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dbf8b880776c6738161e718a19a80cb010d6f8) |
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Arithmetic mean
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![{\displaystyle F[y]=\exp \left({\frac {1}{t-a}}\int _{a}^{t}\ln y\,dt\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f002952416bda45b3935e147ad3dbe8a88423e6) |
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Geometric mean
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![{\displaystyle F[y]=-{\frac {y}{y'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b744a33e05acd24dc342902fc4435f9b98cc02b) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Subtangent
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![{\displaystyle F[x,y]=-{\frac {yx'}{y'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf503f7c4cfe9a01ed4600ce99056a44c0ddcbb5) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
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![{\displaystyle F[r]=-{\frac {r^{2}}{r'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b8c3e99b9a9d37d6709af9ebc75662c4894e1f9) |
Polar |
![{\displaystyle r=r(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088f42375db48e01cbb60551e178e265a5c93b70)
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![{\displaystyle F[r]={\frac {1}{2}}\int _{a}^{t}r^{2}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05bffb44214252eaa080593550c1fadb19f401a1) |
Polar |
![{\displaystyle r=r(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088f42375db48e01cbb60551e178e265a5c93b70)
![{\displaystyle \phi =t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/776cc0bed4150a3698e13afb923444b487075e06) |
Sector area
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![{\displaystyle F[y]=\int _{a}^{t}{\sqrt {1+y'^{2}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f5075734f6fe703a7580c22819a9bdc4fd8628) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Arc length
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![{\displaystyle F[x,y]=\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6ca40e6175c2bb3c7322c838a57f33a2f2ec39) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
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![{\displaystyle F[r]=\int _{a}^{t}{\sqrt {r^{2}+r'^{2}}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c184b7ab8977f6a6edc027a9e1acdada1974ea0) |
Polar |
![{\displaystyle r=r(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088f42375db48e01cbb60551e178e265a5c93b70)
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![{\displaystyle F[y]=\int _{a}^{t}{\sqrt[{3}]{y''}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e29ec01bfa19bfb818eed88beb5eeebea6dbc7d) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Affine arc length
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![{\displaystyle F[x,y]=\int _{a}^{t}{\sqrt[{3}]{x'y''-x''y'}}\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47508e30fcd4d476900960eb4f84b3c1295c0ae7) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
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![{\displaystyle F[x,y,z]=\int _{a}^{t}{\sqrt[{3}]{z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/366fb346e65a188ad374a17d1edc947518cafc87) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1)
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![{\displaystyle F[y]={\frac {y''}{(1+y'^{2})^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2733d361692921c78d8491e12ff0d3975e46672e) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Curvature
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![{\displaystyle F[x,y]={\frac {x'y''-y'x''}{(x'^{2}+y'^{2})^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0092d291416fcd260b739fa7015c28619ee93702) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
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![{\displaystyle F[r]={\frac {r^{2}+2r'^{2}-rr''}{(r^{2}+r'^{2})^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa7d7dbbee040a4d5826ba4ab9f7cb82ce65a18) |
Polar |
![{\displaystyle r=r(\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/088f42375db48e01cbb60551e178e265a5c93b70)
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![{\displaystyle F[x,y,z]={\frac {\sqrt {(z''y'-z'y'')^{2}+(x''z'-z''x')^{2}+(y''x'-x''y')^{2}}}{(x'^{2}+y'^{2}+z'^{2})^{3/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5207fd18045bcdfa79d3df3e0f09fea7d962d443) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1)
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![{\displaystyle F[y]={\frac {1}{3}}{\frac {y''''}{(y'')^{5/3}}}-{\frac {5}{9}}{\frac {y'''^{2}}{(y'')^{8/3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e7a3650eaa04be655adc42c3346eb29358961ee) |
Cartesian |
![{\displaystyle y=y(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66fc1db874a7395b28e294c7db30f51bcd8df20)
![{\displaystyle x=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6c0426ebd38c267805e4308c63f11cb976e4963) |
Affine curvature
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![{\displaystyle F[x,y]={\frac {x''y'''-x'''y''}{(x'y''-x''y')^{5/3}}}-{\frac {1}{2}}\left[{\frac {1}{(x'y''-x''y')^{2/3}}}\right]''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a69d739b8216cae91fad1ca47b9e723093682d8) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
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![{\displaystyle F[x,y,z]={\frac {z'''(x'y''-y'x'')+z''(x'''y'-x'y''')+z'(x''y'''-x'''y'')}{(x'^{2}+y'^{2}+z'^{2})(x''^{2}+y''^{2}+z''^{2})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2173260d52b59f35d1d713a8378669f8e5a494b) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1)
![{\displaystyle z=z(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac2a2dee3e259987e25cb3bd7633903bd154ade) |
Torsion of curves
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![{\displaystyle X[x,y]={\frac {y'}{yx'-xy'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b267261ce0de1ba0a6ca53134308e5a263da24d1)
![{\displaystyle Y[x,y]={\frac {x'}{xy'-yx'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547f7562b2770e9c54b89f99e4b9ad2b1d9ede0f) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Dual curve (tangent coordinates)
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![{\displaystyle X[x,y]=x+{\frac {ay'}{\sqrt {x'^{2}+y'^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7049a1a66ae0e3a174a1953af510094495b5a2e)
![{\displaystyle Y[x,y]=y-{\frac {ax'}{\sqrt {x'^{2}+y'^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b555b5ef64340d7b8110451e067703a405d379) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Parallel curve
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![{\displaystyle X[x,y]=x+y'{\frac {x'^{2}+y'^{2}}{x''y'-y''x'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/863007693180945db748fccf523a900cd172d85d)
![{\displaystyle Y[x,y]=y+x'{\frac {x'^{2}+y'^{2}}{y''x'-x''y'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b73ec650428cdcc8f03cb9444aee9b49b6dcf05) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Evolute
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![{\displaystyle F[r]=t(r'\circ r^{[-1]})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3562fadfb34e7d19384e119e1f01f1d57ee84b2) |
Intrinsic |
![{\displaystyle r=r(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/078f120f6255f798e14a3953003bc402b243fd0b)
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![{\displaystyle X[x,y]=x-{\frac {x'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/203e22b2482eac10aaea0446f9a3e69e82db9674)
![{\displaystyle Y[x,y]=y-{\frac {y'\int _{a}^{t}{\sqrt {x'^{2}+y'^{2}}}\,dt}{\sqrt {x'^{2}+y'^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f2233dc9b36793a05d1afd06e88265b393567b) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Involute
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![{\displaystyle X[x,y]={\frac {(xy'-yx')y'}{x'^{2}+y'^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db71b35d67e2606dcf3a05acce73dc9193767cb6)
![{\displaystyle Y[x,y]={\frac {(yx'-xy')x'}{x'^{2}+y'^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39388d40e20824a70c17f7a32237033623ba7439) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Pedal curvewith pedal point (0;0)
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![{\displaystyle X[x,y]={\frac {(x'^{2}-y'^{2})y'+2xyx'}{xy'-yx'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c924853b4907ed4da8a2ac246d9580e8bc44f8)
![{\displaystyle Y[x,y]={\frac {(x'^{2}-y'^{2})x'+2xyy'}{xy'-yx'}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79fe316c48155e6f92993f1e3c1393c47e30699c) |
Parametric Cartesian |
![{\displaystyle x=x(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae1e02df253e3adc6e5d080f37a40a5bc805320)
![{\displaystyle y=y(t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b87dc4cd066d2bdd5379752e0fd7133eb79c2f1) |
Negative pedal curvewith pedal point (0;0)
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![{\displaystyle X[y]=\int _{a}^{t}\cos \left[\int _{a}^{t}{\frac {1}{y}}\,dt\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/935c78f92c71c500905f0e5eb20097dbffb1f5d0)
![{\displaystyle Y[y]=\int _{a}^{t}\sin \left[\int _{a}^{t}{\frac {1}{y}}\,dt\right]dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed7d68b49a43d3e04780d4f006de948ae7216e1) |
Intrinsic |
![{\displaystyle y=r(s)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8b91117c8bd35734c40c0b007f4d9cbd3b591c)
![{\displaystyle s=t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0fd4cbdea3f3c94e1dd79dee73d87a150dbf3e) |
Intrinsic to Cartesian transformation
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Metric functionals
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![{\displaystyle F[y]=\|y\|={\sqrt {\int _{E}y^{2}\,dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4017443e893dd3adb4e88cc75417d5b01ccc16e0) |
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Norm
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![{\displaystyle F[x,y]=\int _{E}xy\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/802878f8dc672ed389e2eb52e0260cbedbf0f738) |
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Inner product
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![{\displaystyle F[x,y]=\arccos \left[{\frac {\int _{E}xy\,dt}{{\sqrt {\int _{E}x^{2}\,dt}}{\sqrt {\int _{E}y^{2}\,dt}}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab83c107b4f06a04bf00c6b25aea41968897760d) |
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Fubini–Study metric (inner angle)
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Distribution functionals
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![{\displaystyle F[x,y]=x*y=\int _{E}x(s)y(t-s)\,ds}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eacc479d5118751b7e740e1e4383ee523604d3a) |
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Convolution
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![{\displaystyle F[y]=\int _{E}y\ln y\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a14fdd52adf147b44d54644815af4b7a45ef0c58) |
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Differential entropy
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![{\displaystyle F[y]=\int _{E}yt\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c320e540a14d41266a5d973b60b214075ef63b40) |
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Expected value
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![{\displaystyle F[y]=\int _{E}\left(t-\int _{E}yt\,dt\right)^{2}y\,dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1324142b0ca262523919913e4949fe2a0a7be9c5) |
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Variance
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