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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > " " " < / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > 计算三元函数在某点的梯度向量。< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > > [!tip]< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > > 已知一个函数$f(x, y, z)$，则其在点$(x_0, y_0, z_0)$处的梯度向量为:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > $< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > nabla f(x_0, y_0, z_0) = < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > left(< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > frac{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial f}{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial x}, < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > frac{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial f}{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial y}, < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > frac{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial f}{< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > partial z}< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > \\< / span > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > right)$< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > Args:< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > func ([`ThreeSingleVarsFunc`](./mp_math_typing#var-threesinglevarsfunc)): 三元函数< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > p ([`Point3`](./point#class-point3)): 点< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > epsilon: 偏移量< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > Returns:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > 梯度< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > " " " < / span > < / span >
< span class = "line" > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > dx < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > =< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (func(p.x < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > +< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon, p.y, p.z) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > func(p.x < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon, p.y, p.z)) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > /< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 2< / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon)< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > dy < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > =< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (func(p.x, p.y < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > +< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon, p.z) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > func(p.x, p.y < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon, p.z)) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > /< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 2< / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon)< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > dz < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > =< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (func(p.x, p.y, p.z < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > +< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > func(p.x, p.y, p.z < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > -< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon)) < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > /< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 2< / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > epsilon)< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > return< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > Vector3(dx, dy, dz)< / span > < / span > < / code > < / pre > < / div > < / details > < h3 id = "def-curry-func-multivarsfunc-args-var-onevarfunc" tabindex = "-1" > < em > < strong > def< / strong > < / em > < code > curry(func: MultiVarsFunc, *args: Var) -> OneVarFunc< / code > < a class = "header-anchor" href = "#def-curry-func-multivarsfunc-args-var-onevarfunc" aria-label = "Permalink to "***def*** `curry(func: MultiVarsFunc, *args: Var) -> OneVarFunc`"" > < / a > < / h3 > < p > < strong > 说明< / strong > : 对多参数函数进行柯里化。< / p > < div class = "tip custom-block github-alert" > < p class = "custom-block-title" > TIP< / p > < p > 有关函数柯里化，可参考< a href = "https://zhuanlan.zhihu.com/p/355859667" target = "_blank" rel = "noreferrer" > 函数式编程--柯理化（ ） < / a > < / p > < / div > < p > < strong > 参数< / strong > :< / p > < blockquote > < ul > < li > func (< a href = "./mp_math_typing.html#var-multivarsfunc" > < code > MultiVarsFunc< / code > < / a > ): 函数< / li > < li > *args (< a href = "./mp_math_typing.html#var-var" > < code > Var< / code > < / a > ): 参数< / li > < / ul > < / blockquote > < p > < strong > 返回< / strong > : 柯里化后的函数< / p > < p > < strong > 示例< / strong > :< / p > < div class = "language-python vp-adaptive-theme" > < button title = "Copy Code" class = "copy" > < / button > < span class = "lang" > python< / span > < pre class = "shiki shiki-themes github-light github-dark vp-code" tabindex = "0" > < code > < span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > def< / span > < span style = "--shiki-light:#6F42C1;--shiki-dark:#B392F0;" > add< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (a: < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > int< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > , b: < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > int< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > , c: < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > int< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > ) -> < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > int< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > :< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > return< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > a < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > +< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > b < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > +< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > c< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > add_curried < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > =< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > curry(add, < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 1< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > , < / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 2< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > )< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > add_curried(< / span > < span style = "--shiki-light:#005CC5;--shiki-dark:#79B8FF;" > 3< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > ) < / span > < span style = "--shiki-light:#6A737D;--shiki-dark:#6A737D;" > # 6< / span > < / span > < / code > < / pre > < / div > < details > < summary > < b > 源代码< / b > 或 < a href = "https://github.com/snowykami/mbcp/tree/main/mbcp/mp_math/function.py#L30" target = "_blank" > 在GitHub上查看< / a > < / summary > < div class = "language-python vp-adaptive-theme" > < button title = "Copy Code" class = "copy" > < / button > < span class = "lang" > python< / span > < pre class = "shiki shiki-themes github-light github-dark vp-code" tabindex = "0" > < code > < span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > def< / span > < span style = "--shiki-light:#6F42C1;--shiki-dark:#B392F0;" > curry< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (func: MultiVarsFunc, < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > args: Var) -> OneVarFunc:< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > " " " < / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > 对多参数函数进行柯里化。< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > > [!tip]< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > > 有关函数柯里化，可参考[函数式编程--柯理化（ ） < / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > Args:< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > func ([`MultiVarsFunc`](./mp_math_typing#var-multivarsfunc)): 函数< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > *args ([`Var`](./mp_math_typing#var-var)): 参数< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > Returns:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > 柯里化后的函数< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > Examples:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > ```python< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > def add(a: int, b: int, c: int) -> int:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > return a + b + c< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > add_curried = curry(add, 1, 2)< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > add_curried(3) # 6< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > ```< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > " " " < / span > < / span >
< span class = "line" > < / span >
< span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > def< / span > < span style = "--shiki-light:#6F42C1;--shiki-dark:#B392F0;" > curried_func< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > (< / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > args2: Var) -> Var:< / span > < / span >
< span class = "line" > < span style = "--shiki-light:#032F62;--shiki-dark:#9ECBFF;" > " " " @litedoc-hide" " " < / span > < / span >
< span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > return< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > func(< / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > args, < / span > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > *< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > args2)< / span > < / span >
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< span class = "line" > < span style = "--shiki-light:#D73A49;--shiki-dark:#F97583;" > return< / span > < span style = "--shiki-light:#24292E;--shiki-dark:#E1E4E8;" > curried_func< / span > < / span > < / code > < / pre > < / div > < / details > < / div > < / div > < / main > < footer class = "VPDocFooter" data-v-40342069 data-v-a4b38bd6 > <!-- [ --> <!-- ] --> < div class = "edit-info" data-v-a4b38bd6 > < div class = "edit-link" data-v-a4b38bd6 > < a class = "VPLink link vp-external-link-icon no-icon edit-link-button" href = "https://github.com/snowykami/mbcp/tree/main/mbcp//mp_math/function.py" target = "_blank" rel = "noreferrer" data-v-a4b38bd6 > <!-- [ --> < span class = "vpi-square-pen edit-link-icon" data-v-a4b38bd6 > < / span > 在 GitHub 上编辑此页<!-- ] --> < / a > < / div > <!-- --> < / div > < nav class = "prev-next" aria-labelledby = "doc-footer-aria-label" data-v-a4b38bd6 > < span class = "visually-hidden" id = "doc-footer-aria-label" data-v-a4b38bd6 > Pager< / span > < div class = "pager" data-v-a4b38bd6 > < a class = "VPLink link pager-link prev" href = "/api/mp_math/equation.html" data-v-a4b38bd6 > <!-- [ --> < span class = "desc" data-v-a4b38bd6 > 上一页< / span > < span class = "title" data-v-a4b38bd6 > mbcp.mp_math.equation< / span > <!-- ] --> < / a > < / div > < div class = "pager" data-v-a4b38bd6 > < a class = "VPLink link pager-link next" href = "/api/mp_math/" data-v-a4b38bd6 > <!-- [ --> < span class = "desc" data-v-a4b38bd6 > 下一页< / span > < span class = "title" data-v-a4b38bd6 > mbcp.mp_math< / span > <!-- ] --> < / a > < / div > < / nav > < / footer > <!-- [ --> <!-- ] --> < / div > < / div > < / div > <!-- [ --> <!-- ] --> < / div > < / div > < footer class = "VPFooter has-sidebar" data-v-22f859ac data-v-e3ca6860 > < div class = "container" data-v-e3ca6860 > < p class = "message" data-v-e3ca6860 > 文档由 < a href = "https://vitepress.dev/" > VitePress< / a > 构建 | API引用由 < a href = "https://github.com/LiteyukiStudio/litedoc" > litedoc< / a > 生成< / p > < p class = "copyright" data-v-e3ca6860 > Copyright (C) 2020-2024 SnowyKami. All Rights Reserved< / p > < / div > < / footer > <!-- [ --> <!-- ] --> < / div > < / div >
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< script > w i n d o w . _ _ V P _ H A S H _ M A P _ _ = J S O N . p a r s e ( " { \ " a p i _ a p i . m d \ " : \ " r n P O v 6 - O \ " , \ " a p i _ i n d e x . m d \ " : \ " q n r S d _ _ i \ " , \ " a p i _ m p _ m a t h _ a n g l e . m d \ " : \ " 3 N a 5 I a Y p \ " , \ " a p i _ m p _ m a t h _ c o n s t . m d \ " : \ " O 6 c T u l w X \ " , \ " a p i _ m p _ m a t h _ e q u a t i o n . m d \ " : \ " z 8 5 z y C Z L \ " , \ " a p i _ m p _ m a t h _ f u n c t i o n . m d \ " : \ " B h m J q p O o \ " , \ " a p i _ m p _ m a t h _ i n d e x . m d \ " : \ " D H h P N V 6 R \ " , \ " a p i _ m p _ m a t h _ l i n e . m d \ " : \ " D o v i 6 b d M \ " , \ " a p i _ m p _ m a t h _ m p _ m a t h . m d \ " : \ " m V a b Z T H d \ " , \ " a p i _ m p _ m a t h _ m p _ m a t h _ t y p i n g . m d \ " : \ " C E r y C R E V \ " , \ " a p i _ m p _ m a t h _ p l a n e . m d \ " : \ " D v - G W b J g \ " , \ " a p i _ m p _ m a t h _ p o i n t . m d \ " : \ " B v v r K T Y 8 \ " , \ " a p i _ m p _ m a t h _ s e g m e n t . m d \ " : \ " _ n _ _ V i V t \ " , \ " a p i _ m p _ m a t h _ u t i l s . m d \ " : \ " k W 3 A - 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p r a c t i c e . m d \ " : \ " C P N b D _
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