The Gâteaux differential , named after René Gâteaux (1889–1914), represents a generalization of the common notion of differentiation in that it defines the directional derivation also in infinite-dimensional spaces. Usually one has an open set for a function that is differentiable at that point as the definition of the partial derivative
![{\ displaystyle f \ colon G \ to \ mathbb {R}, \ G \ subset \ mathbb {R} ^ {n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa805a88fe73340fe5c6fc6048e095730ddec5ee)
![x_ {0} \ in G](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d692c82266244320ae3e170b707871569ece64d)
-
.
In particular, it results for the known differential
![n = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425)
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.
The Gâteaux differential generalizes these concepts to infinite-dimensional vector spaces.
Definitions
Weierstrasse's decomposition formula
Be with open and standardized spaces. Then in Gâteaux is called-differentiable if the Weierstrasse decomposition formula holds, i.e. if a linear function exists such that
![f \ colon D \ subset X \ to Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/6213dc6d28ac19ed8b2ec568a5e25e8637609b36)
![D.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6)
![X, Y](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8705438171d938b7f59cd1bfa5b7d99b6afa5cd)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![x_ {0} \ in D](https://wikimedia.org/api/rest_v1/media/math/render/svg/b107b130543b591ac3037f4bc152da314ec51586)
![A \ in L (X, Y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cbe0791c75a545545981ea5f37e9b5592f8a697)
![\ lim _ {{t \ rightarrow 0}} {\ frac {1} {t}} [f (x_ {0} + th) -f (x_ {0}) - tAh] = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9e746fd10e5d7c59bf6e10b96911f0a5cfc10f)
for everyone with . This is equivalent to:
![h \ in X](https://wikimedia.org/api/rest_v1/media/math/render/svg/60183524e60181f3f0fa2b09cd4c27efc3bcca39)
![\ lVert h \ rVert = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/71ede2f666caa783f95866a630ee7bd68d3d33c5)
![f (x_ {0} + th) -f (x_ {0}) = tAh + o (| t |)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e311dd69b4cede3b9258e13b22b258f32e20d46e)
Then it is called the Gâteaux derivation of in the point .
![A =: f '(x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2714c6be93bab5ce60d44d8f61850e91a1c8c48)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
1. variation; Variation derivative
Let the following situation be given for the Gâteaux differential: Let it be, as usual,
a functional defined in; be a linear normed space (that is, a vector space , provided with a norm ) or a more general topological vector space with prerequisites, about which one has to think more closely in the concrete application; further be and . Then the Gâteaux differential is at the point in direction , if it exists, defined by the following derivative :
![f \ colon D (f) \ to {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc2dde254243a74e809b3fbdb6156b5deb7b4f7)
![D (f) \ subseteq \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e7c199bd0d6003faed04142cafbe498b5811147)
![\ | \ cdot \ |](https://wikimedia.org/api/rest_v1/media/math/render/svg/113f0d8fe6108fc1c5e9802f7c3f634f5480b3d1)
![x_ {0} \ in D (f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6134dd7b7b85a7128b55bcef35c75b10a5df630)
![v \ in \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/0641a0e2d18ca75ae13c3fbc0d01daf63c6cbbc6)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![\ delta f (x_ {0}, v) = \ lim _ {{\ epsilon \ to 0}} {\ frac {f (x_ {0} + \ epsilon \ cdot v) -f (x_ {0})} {\ epsilon}} = \ left. {\ frac {{\ mathrm {d}} f (x_ {0} + \ epsilon \ cdot v)} {{\ mathrm {d}} \ epsilon}} \ right | _ {{\ epsilon = 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/566b48665408191b4346d280485e167cb902a6b4)
or for by
![x_ {1} \ in D (f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d841b6f2757be032f94e85d1b328a302c695e32e)
![\ delta f (x_ {0}, x_ {1} -x_ {0}) = \ lim _ {{\ epsilon \ to 0}} {\ frac {f (x_ {0} + \ epsilon \ cdot (x_ { 1} -x_ {0})) - f (x_ {0})} {\ epsilon}} \,. \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/75ee2e57832875a2b6c49c181dec03866c488efc)
Note here , and also in it, but .
![x_ {0} \ in D (f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6134dd7b7b85a7128b55bcef35c75b10a5df630)
![v \ in \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/0641a0e2d18ca75ae13c3fbc0d01daf63c6cbbc6)
![x_ {1} -x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2be377fc132914a5a9e9a747065b89768311e0b)
![\ epsilon \ in {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7125392cc5989f0ffbcfc9096f0c67b5a1e6cf12)
The Gâteaux derivation according to is a functional with regard to size , which is also referred to as the 1st variation of at this point .
![\epsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2)
![h: = x_ {1} -x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbdf1fab63daacaa0a44fd454b5175c39d5f484)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
Another possibility is to use more general topological vector spaces with a corresponding concept of convergence instead of normalized vector spaces . In physics books in particular, functionals are usually denoted by letters , and instead of size one usually writes with distribution-valued quantities. Instead of the derivation , the so-called variation derivation is introduced in an additional step , which is closely related to the Gâteaux derivation.
![I.](https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f)
![h: = x_ {1} -x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbdf1fab63daacaa0a44fd454b5175c39d5f484)
![\ delta q (x)](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0cee3c7b1cfb2e6b1ade01f86251f401a7e035)
![{\ tfrac {dI (x + \ epsilon \ cdot h)} {d \ epsilon}} _ {{\, | \ epsilon = 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/390fe32140cb08d16431dff8259df37046b030ab)
example
For
![f (\ epsilon): = \ int \, {{\ rm {d}}} t \, {\ mathcal L} \ left (t, q (t) + \ epsilon \ cdot \ delta q (t), { \ dot q} (t) + \ epsilon \ cdot {\ frac {{{\ rm {d}}} (\ delta q (t))} {{{\ rm {d}}} t}} \ right)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6edab4f8adbe30b781f49b39b5ddd456170fa25)
after a partial integration with a vanishing, fully integrated part, a result of the form with the derivative of the variation is obtained
![{\ frac {\ delta {\ mathcal L}} {\ delta q (t)}} \ equiv {\ frac {\ partial {\ mathcal L}} {\ partial q (t)}} - {\ frac {{ {\ rm {d}}}} {{{\ rm {d}}} t}} {\ frac {\ partial {\ mathcal L}} {\ partial {\ dot q} (t)}} \ ,.](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf93f14f7071875ac5ffd1b43471e88ccc187e29)
(The variation derivative "at point q (t)" for continuous variables is therefore the generalization of the partial derivative of a function from n variables, for example for the fictitious case . Similar to the total differential of a function of n variables in the fictitious case , so here , too , the total differential of the functional has an invariant meaning. Further details in the chapter Lagrange formalism .)
![{\ tfrac {\ partial {\ mathcal L}} {\ partial x_ {i}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b49eda063d5da403dc1c2af323586c3f22a85e)
![{\ mathcal L} = {\ mathcal L} (x_ {1}, ..., x_ {n})](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9f66d10b69bc8dc850c5cf70ce0931d73bbc433)
![\ delta f](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c10972e3d8cc0def199b8f39ad05bb75ddfdc00)
In the following, for the sake of simplicity, the vectors are not identified by "bold" letters.
2nd variation
![\ delta ^ {2} f (x_ {0}, v) = \ left. {\ frac {{\ mathrm {d}} ^ {2} f (x_ {0} + \ epsilon \ cdot v)} {{ \ mathrm {d}} \ epsilon ^ {2}}} \ right | _ {{\ epsilon = 0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3be47373ba3a95826bf1ee9c5aac7e2067dbd6fa)
Half-sided differential and directional derivative
Under the same conditions as above, the one-sided Gâteaux differential is through
![\ delta _ {+} f (x_ {0}, v) = \ lim _ {{\ epsilon \ to 0 ^ {+}}} {\ frac {f (x_ {0} + \ epsilon \ cdot v) - f (x_ {0})} {\ epsilon}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f13cdc3047ffd0121b36e2e328e8e604fd00b4d)
respectively through
![\ delta _ {-} f (x_ {0}, v) = \ lim _ {{\ epsilon \ to 0 ^ {-}}} {\ frac {f (x_ {0} + \ epsilon \ cdot v) - f (x_ {0})} {\ epsilon}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef1f8951990b921630b748200009074f9fc2a95)
Are defined. The one-sided Gâteaux differential is also called the directional differential of at the point . For the direction belonging to the vector , in the case of “continuous variables”, the one-sided Gâteaux differential (more precisely: the associated variation derivative) just generalizes the directional derivative from in the direction at the point .
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
Gâteaux derivative
If an in continuous linear functional (i.e. the function mediated by is homogeneous, additive and continuous in the argument ), then it is called Gâteaux-derivative in that place and Gâteaux-differentiable in .
![\ delta f (x_ {0}, v)](https://wikimedia.org/api/rest_v1/media/math/render/svg/affb68c8ef803f4e70c8bc72bda51bd29b041d3e)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![v \ mapsto \ delta f (x_ {0}, v)](https://wikimedia.org/api/rest_v1/media/math/render/svg/41bf50336a72bd83071524451264dc0ce5286c15)
![v](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
![f '(x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc15f7bc4034ace9faccf92eb8e3f245541c5e6e)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
Properties of the 1st variation
- The Gâteaux differential is homogeneous, which means for everyone . The same applies to the one-sided Gâteaux differential.
![\ delta f (x_ {0}, k \ cdot v) = k \ cdot \ delta f (x_ {0}, v)](https://wikimedia.org/api/rest_v1/media/math/render/svg/f705e3c48516811ae353f509b77d187707653a38)
![k \ in {\ mathbb {R}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/177754a3bb6c26dbd54cbc866337b20bafa64e7c)
- The Gâteaux differential is not linear. In the general case, for an example that the Gâteaux differential is not linear, consider for and , where , then is . The function is not linear. It applies, for example .
![\ delta f (x_ {0}, v) + \ delta f (x_ {0}, w) \ neq \ delta f (x_ {0}, v + w).](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3b1dccf62b3a282565ff415eed643d7364da9b)
![f (x) = {\ tfrac {x_ {1} ^ {2} x_ {2}} {x_ {1} ^ {2} + x_ {2} ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed12feb23d90e363161f9f545f09a313fd7dd08e)
![x \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/35a455db7b2aab1b0e72ccbc7385e4424e2372e5)
![f (0) = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d308c32c9894b88115262081194321ae7d9bbf3)
![x = (x_1, x_2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd63ad6de0b5e6df770b048ea4ba977d1ff1115)
![\ delta f (0, v) = \ lim _ {{\ epsilon \ to 0}} {\ frac {f (0+ \ epsilon \ cdot v) -f (0)} {\ epsilon}} = \ lim _ {{\ epsilon \ to 0}} {\ frac {f (\ epsilon \ cdot v)} {\ epsilon}} = \ lim _ {{\ epsilon \ to 0}} {\ frac {\ epsilon ^ {2} v_ {1} ^ {2} \ cdot \ epsilon v_ {2}} {(\ epsilon ^ {2} v_ {1} ^ {2} + \ epsilon ^ {2} v_ {2} ^ {2}) \ epsilon}} = {\ frac {v_ {1} ^ {2} v_ {2}} {v_ {1} ^ {2} + v_ {2} ^ {2}}} = f (v)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0353c181fcabd1c2e4e063e10c53cd97932479b)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ delta f (0, (0.1)) + \ delta f (0, (1.0)) = 0 + 0 \ neq {\ tfrac {1} {2}} = \ delta f (0, (1 ,1))](https://wikimedia.org/api/rest_v1/media/math/render/svg/05d806f6d7bbd23851819947a28043b0dcad03d5)
Examples
-
if , or otherwise .![x_ {2} = x_ {1} ^ {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a178941019594873ab579f8a9fd70594867cb62)
![x_ {1} \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a5419a8f07b0d246f0002346520867a0ff152a)
![{\ displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950)
![\ delta f ((0,0), v) = \ lim _ {{t \ to 0}} {\ frac {0-0} {t}} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/3985657177595e6bb6e9d108b4607f7835f76c5d)
-
-
for and for ,![x_ {2} \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0e0b0b810812acd81e12743fc8b070f7a641c08)
![- {\ frac {x_ {1} ^ {2}} {x_ {2} ^ {2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29660c68e093b121473707e5ec2457a2c454a087)
![x_2 = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/1728b541c2d859bd489c1225a6aae0452099e611)
(where )
![v = (v_ {1}, v_ {2}) ^ {{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e80ac0586cb339e4f359e6fffaf6893400edc92)
Applications
Like the usual derivative , the Gâteaux differential is useful for determining extrema and therefore in optimization . Be open, linear normalized space, (the interior of the set ), and the open ball around with radius . Necessary optimality condition: Let be a local minimum of on , then if the one-sided Gâteaux differential in exists. Sufficient optimality condition: have in a 2nd variation and . If and for one and , then there is a strict local minimum of on .
![{\ displaystyle f \ colon X \ to \ mathbb {R}, \ X \ subset D (f) \ subset \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e483bb8e5b393169eb493c3c7d2a71c2da0cbf9)
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
![x_ {0} \ in \ operatorname {int} (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6992d53aab999fad622068385375789240583e85)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![\ operatorname {int} (X) \ neq \ emptyset](https://wikimedia.org/api/rest_v1/media/math/render/svg/47f496d23ab9385c60104ed06306b74a48972b25)
![B _ {\ varepsilon} (x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d81ce8a55aaa9019de68c5e956ef698f894ad0f)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![\ varepsilon](https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![X](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
![{\ displaystyle \ delta _ {+} f (x_ {0}, v) \ geq 0 \ \ forall v \ in \ Omega}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21322dd798b1908899f21267234e4653e39071fb)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![B _ {\ varepsilon} (x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d81ce8a55aaa9019de68c5e956ef698f894ad0f)
![\ forall v \ in \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/78f810247fe9e3240b1b558d8a93f91e1346f31d)
![\ forall x \ in B _ {\ varepsilon} (x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f44b888b4413176da04ff47a0ae767931e21de)
![\ delta f (x_ {0}, v) = 0 \ \ forall v \ in \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0d1dde4d58127428c4c1382fc3c7fa81ceb39a)
![\ delta ^ {2} f (x_ {0}, v) \ geq c \ cdot \ | v \ | ^ {2} \ \ forall v \ in \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcb8f83914bbc99ece0ef7aa6afbad26dc4544b7)
![\ forall x \ in B _ {\ varepsilon} (x_ {0})](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f44b888b4413176da04ff47a0ae767931e21de)
![x_ {0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf)
![f](https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61)
![\ operatorname {int} (X)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a18947840e2d885fde9c2340adf6904d391651c3)
See also