在数学和理论物理中,泛函导数是方向导数的推广。后者对一个有限维向量求微分,而前者则对一个连续函数(可视为无穷维向量)求微分。它们都可以认为是简单的一元微积分中导数的扩展。数学里专门研究泛函导数的分支是泛函分析。
设有流形 M 代表(连续/光滑/有某些边界条件等的)函数 φ 以及泛函 F:
,
则F的泛函导数,记为
,是一个满足以下条件的分布:
对任何测量函数 f:
用
的一次变分
代替
就得到
的一次变分
;
在物理学中,通常用狄拉克δ函数
,而不是一般的测试函数
, 来求出点
处的泛函导数(这是整个泛函变分的关键点,就像偏导数是梯度的一个分量):
![{\displaystyle {\frac {\delta F[\varphi (x)]}{\delta \varphi (y)}}=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon \delta (x-y)]-F[\varphi (x)]}{\varepsilon }}.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0d1f0e4f15b7bd5f256badc12ab90cf6d7740d11)
这适用于
可以展开成
的级数時 (或者至少能展为1阶). 但是这一表达在数学上并不严格,因为
一般而言并未定义。
通过更仔细地定义函数空间,泛函导数的定义可以更准确、正式。例如,当函数空间是一个巴拿赫空间时, 泛函导数就是著名的Fréchet导数, 而这在更一般的局部凸空间上使用加托導數。注意,著名的希尔伯特空间是巴拿赫空间的特例。更正式的处理允许将普通微积分和数学分析的定理推广为泛函分析中对应的定理,以及大量的新定理。
與函數的導數類似,泛函導數滿足下列的性質:(其中 F[ρ] 和 G[ρ] 為兩個泛函)
![{\displaystyle {\frac {\delta (\lambda F+\mu G)[\rho ]}{\delta \rho (x)}}=\lambda {\frac {\delta F[\rho ]}{\delta \rho (x)}}+\mu {\frac {\delta G[\rho ]}{\delta \rho (x)}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a3855d2be1b75c59c9da233cf9c8c71eeed66eaf)
- 其中 λ, μ 皆為常數。
![{\displaystyle {\frac {\delta (FG)[\rho ]}{\delta \rho (x)}}={\frac {\delta F[\rho ]}{\delta \rho (x)}}G[\rho ]+F[\rho ]{\frac {\delta G[\rho ]}{\delta \rho (x)}}\,,}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ee0421ecdfb105f4b62dcbd729c20e8ca587bda2)
- 若 F 和 G 為兩個泛函,則[3]
![{\displaystyle \displaystyle {\frac {\delta F[G[\rho ]]}{\delta \rho (y)}}=\int dx{\frac {\delta F[G(\rho )]}{\delta G[\rho (x)]}}\ {\frac {\delta G[\rho ]}{\delta \rho (y)}}\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/216041b2b3d01f08208dc647d3eee711c813b7bc)
- 若當中的 G 為一個普通的可導函數 g,則上式化為[4]
![{\displaystyle \displaystyle {\frac {\delta F[g(\rho )]}{\delta \rho (y)}}={\frac {\delta F[g(\rho )]}{\delta g[\rho (x)]}}\ {\frac {dg(\rho )}{d\rho (y)}}\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1433a84901f009da314d7e7a3cf26d3ff027aa87)
上面给出的定义是基于一种对所有测量函数 f都成立的关系,因此有人可能会想,它在 f是一个指定的函数(比如说狄拉克δ函数)时也应该成立。但是,δ函数不是一个合理的测量函数。
在定义中,泛函导数描述了整个函数
发生微小变化时,泛函
如何变化。其中,
的变化量的具体形式没有指明,
給定泛函
![{\displaystyle F[\rho ]=\int f({\boldsymbol {r}},\rho ({\boldsymbol {r}}),\nabla \rho ({\boldsymbol {r}}))\,d{\boldsymbol {r}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/984500e18fa20fadb9f03147b92b046a166aafc7)
及在積分區域的邊界上恆為零的函數 ϕ(r),由定義可得:
![{\displaystyle {\begin{aligned}\int {\frac {\delta F}{\delta \rho ({\boldsymbol {r}})}}\,\phi ({\boldsymbol {r}})\,d{\boldsymbol {r}}&=\left[{\frac {d}{d\varepsilon }}\int f({\boldsymbol {r}},\rho +\varepsilon \phi ,\nabla \rho +\varepsilon \nabla \phi )\,d{\boldsymbol {r}}\right]_{\varepsilon =0}\\&=\int \left({\frac {\partial f}{\partial \rho }}\,\phi +{\frac {\partial f}{\partial \nabla \rho }}\cdot \nabla \phi \right)d{\boldsymbol {r}}\\&=\int \left[{\frac {\partial f}{\partial \rho }}\,\phi +\nabla \cdot \left({\frac {\partial f}{\partial \nabla \rho }}\,\phi \right)-\left(\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi \right]d{\boldsymbol {r}}\\&=\int \left[{\frac {\partial f}{\partial \rho }}\,\phi -\left(\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi \right]d{\boldsymbol {r}}\\&=\int \left({\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}\right)\phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}\,.\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/364480cb1cb7bbad967750d4f4c2b2baa061f134)
其中第二行用到了 f 的全微分, ∂f /∂∇ρ 為純量對向量的導數。[Note 1] 第三行則用到了散度的積法則。第四行由高斯散度定理及邊界上 ϕ=0 的條件得到。由於 ϕ 可以是任意的函數,由變分法基本引理可知,所求泛函導數為
![{\displaystyle {\frac {\delta F}{\delta \rho ({\boldsymbol {r}})}}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial \nabla \rho }}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/22ee5ae13b023dfb79c4ce74ae1c462d0df9b8c0)
其中 ρ = ρ(r) 且 f = f (r, ρ, ∇ρ)。只要 F[ρ] 具有本節首段的形式,上述公式就適用。對於其他的泛函形式,可由定義出發,求出其泛函導數。(見库仑势能泛函。)
以上公式可推廣到高維,並且有其他高階導數的情況。則泛函可寫成
![{\displaystyle F[\rho ({\boldsymbol {r}})]=\int f({\boldsymbol {r}},\rho ({\boldsymbol {r}}),\nabla \rho ({\boldsymbol {r}}),\nabla ^{(2)}\rho ({\boldsymbol {r}}),\dots ,\nabla ^{(N)}\rho ({\boldsymbol {r}}))\,d{\boldsymbol {r}},}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e8d2a74e6ffc8d130e6540ae052afa4431152535)
其中向量 r ∈ ℝn,而 ∇(i) 為一個張量,其 ni 個分量分別為 i 階微分算子
[Note 2]
與上面類似,由泛函導數的定義可知:
![{\displaystyle {\begin{aligned}{\frac {\delta F[\rho ]}{\delta \rho }}&{}={\frac {\partial f}{\partial \rho }}-\nabla \cdot {\frac {\partial f}{\partial (\nabla \rho )}}+\nabla ^{(2)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(2)}\rho \right)}}+\dots +(-1)^{N}\nabla ^{(N)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(N)}\rho \right)}}\\&{}={\frac {\partial f}{\partial \rho }}+\sum _{i=1}^{N}(-1)^{i}\nabla ^{(i)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}\ .\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/5e309ee6857a3699070ebbbb3e9380ededa6572d)
式中,張量
具有 ni 個分量,各為 f 對 ρ 偏導數之偏導數,即:
![{\displaystyle \left[{\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}\right]_{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}={\frac {\partial f}{\partial \rho _{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}}}\qquad \qquad {\text{where}}\quad \rho _{\alpha _{1}\alpha _{2}\cdots \alpha _{i}}\equiv {\frac {\partial ^{\,i}\rho }{\partial r_{\alpha _{1}}\,\partial r_{\alpha _{2}}\cdots \partial r_{\alpha _{i}}}}\ ,}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/75a055a108cb726caf3543357d6e379254a09dbf)
並定義張量的純量積為
[Note 3]
1927年的Thomas-Fermi模型对于无相互作用的单一电子雲使用了动能泛函是密度泛函理论关于电子结构的第一次尝试
![{\displaystyle T_{\mathrm {TF} }[\rho ]=C_{\mathrm {F} }\int \rho ^{5/3}(\mathbf {r} )\,d\mathbf {r} .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0ae24077d52afacfbb4b9b0595c7dcf16977936a)
只与电子密度有关
并且不依赖于其梯度, Laplacian, 或者其他更高阶的微分 (像这样的泛函被称为是“局部的”). 因此,
![{\displaystyle {\frac {\delta T_{\mathrm {TF} }[\rho ]}{\delta \rho }}=C_{\mathrm {F} }{\frac {\partial \rho ^{5/3}(\mathbf {r} )}{\partial \rho (\mathbf {r} )}}={\frac {5}{3}}C_{\mathrm {F} }\rho ^{2/3}(\mathbf {r} ).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4515d2b1b1c2d7a21db603863c296ab28ad29fdb)
托馬斯和費米利用了以下库仑勢能泛函來描述電子與核之間的電勢
![{\displaystyle V[\rho ]=\int {\frac {\rho ({\boldsymbol {r}})}{|{\boldsymbol {r}}|}}\ d{\boldsymbol {r}}.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2c1593ae52b426aa72244fda7d98ac6aab5a6fd4)
由泛函導數的定義,
![{\displaystyle {\begin{aligned}\int {\frac {\delta V}{\delta \rho ({\boldsymbol {r}})}}\ \phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}&{}=\left[{\frac {d}{d\varepsilon }}\int {\frac {\rho ({\boldsymbol {r}})+\varepsilon \phi ({\boldsymbol {r}})}{|{\boldsymbol {r}}|}}\ d{\boldsymbol {r}}\right]_{\varepsilon =0}\\&{}=\int {\frac {1}{|{\boldsymbol {r}}|}}\,\phi ({\boldsymbol {r}})\ d{\boldsymbol {r}}\,.\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c5feda7981e551ab3ef736601c97d676b5c93eeb)
故
![{\displaystyle {\frac {\delta V}{\delta \rho ({\boldsymbol {r}})}}={\frac {1}{|{\boldsymbol {r}}|}}\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/1c1cee60ee7e8ff1b8ce269836ddb38a45629fd6)
至於電子與電子間的相互作用,由以下庫侖勢能泛函描述:
![{\displaystyle J[\rho ]={\frac {1}{2}}\iint {\frac {\rho (\mathbf {r} )\rho (\mathbf {r} ')}{\vert \mathbf {r} -\mathbf {r} '\vert }}\,d\mathbf {r} d\mathbf {r} '\,.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2a6d16f5ade6b1bc0c8c78a03ff33b386221d068)
由定義,
![{\displaystyle {\begin{aligned}\int {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}\phi ({\boldsymbol {r}})d{\boldsymbol {r}}&{}=\left[{\frac {d\ }{d\epsilon }}\,J[\rho +\epsilon \phi ]\right]_{\epsilon =0}\\&{}=\left[{\frac {d\ }{d\epsilon }}\,\left({\frac {1}{2}}\iint {\frac {[\rho ({\boldsymbol {r}})+\epsilon \phi ({\boldsymbol {r}})]\,[\rho ({\boldsymbol {r}}')+\epsilon \phi ({\boldsymbol {r}}')]}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\right)\right]_{\epsilon =0}\\&{}={\frac {1}{2}}\iint {\frac {\rho ({\boldsymbol {r}}')\phi ({\boldsymbol {r}})}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'+{\frac {1}{2}}\iint {\frac {\rho ({\boldsymbol {r}})\phi ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}\,d{\boldsymbol {r}}d{\boldsymbol {r}}'\\\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/604d9921f18f63a32115a37490f300325c7fafcb)
式末的兩個積分相等,因為可以交換第二個積分中 r 和 r′ 兩個變數,而不改變積分的值。因此,
![{\displaystyle \int {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}\phi ({\boldsymbol {r}})d{\boldsymbol {r}}=\int \left(\int {\frac {\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'\right)\phi ({\boldsymbol {r}})d{\boldsymbol {r}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/a37ec62be367efd27e812fb1168b1dca8d1f27ee)
故電子-電子庫侖勢能泛函 J[ρ] 的導數為[5]
![{\displaystyle {\frac {\delta J}{\delta \rho ({\boldsymbol {r}})}}=\int {\frac {\rho ({\boldsymbol {r}}')}{\vert {\boldsymbol {r}}-{\boldsymbol {r}}'\vert }}d{\boldsymbol {r}}'\,.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/f39bc784298e5fd4f7f7503aeff0c0638d2621ad)
且其二階泛函導數為
![{\displaystyle {\frac {\delta ^{2}J[\rho ]}{\delta \rho (\mathbf {r} ')\delta \rho (\mathbf {r} )}}={\frac {\partial }{\partial \rho (\mathbf {r} ')}}\left({\frac {\rho (\mathbf {r} ')}{\vert \mathbf {r} -\mathbf {r} '\vert }}\right)={\frac {1}{\vert \mathbf {r} -\mathbf {r} '\vert }}.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/2e5d2633d38d32e3ec5d142181697543d6b17aea)
1935 年,魏茨泽克提出,在托馬斯-費米動能泛函中添加一項梯度修正,使之能更準確描述分子的電子雲:
![{\displaystyle T_{\mathrm {W} }[\rho ]={\frac {1}{8}}\int {\frac {\nabla \rho (\mathbf {r} )\cdot \nabla \rho (\mathbf {r} )}{\rho (\mathbf {r} )}}d\mathbf {r} =\int t_{\mathrm {W} }\ d\mathbf {r} \,,}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0eafe31ae78208f4c75df2c3147f18c61ed02e29)
其中
![{\displaystyle t_{\mathrm {W} }\equiv {\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\qquad {\text{and}}\ \ \rho =\rho ({\boldsymbol {r}})\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/6ba8b6c7fe84535137fe832d2b552d3ee4d237f0)
由上節的公式可得
![{\displaystyle {\begin{aligned}{\frac {\delta T_{\mathrm {W} }}{\delta \rho ({\boldsymbol {r}})}}&={\frac {\partial t_{\mathrm {W} }}{\partial \rho }}-\nabla \cdot {\frac {\partial t_{\mathrm {W} }}{\partial \nabla \rho }}\\&=-{\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}-\left({\frac {1}{4}}{\frac {\nabla ^{2}\rho }{\rho }}-{\frac {1}{4}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}\right)\qquad {\text{where}}\ \ \nabla ^{2}=\nabla \cdot \nabla \ ,\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7606a5a837e42b5c6fef0dc4f579708e225a4fce)
故所求泛函導數為[6]
![{\displaystyle {\frac {\delta T_{\mathrm {W} }}{\delta \rho ({\boldsymbol {r}})}}=\ \ \,{\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho ^{2}}}-{\frac {1}{4}}{\frac {\nabla ^{2}\rho }{\rho }}\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/baa84f1f6ef2c83d9e6f447b2c48ccdd01430759)
最后,注意到任何函数都可以以积分的形式表示成一个泛函。例如,
![{\displaystyle \rho (\mathbf {r} )=\int \rho (\mathbf {r} ')\delta (\mathbf {r} -\mathbf {r} ')\,d\mathbf {r} '.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/7f67f978e573cf64342eed81cd0ceea9a62a5366)
这个泛函只依赖于
,像上面两个例子一样(就是说,它们都是“局部的”)。因此
![{\displaystyle {\frac {\delta \rho (\mathbf {r} )}{\delta \rho (\mathbf {r} ')}}={\frac {\partial \rho (\mathbf {r} ')\delta (\mathbf {r} -\mathbf {r} ')}{\partial \rho (\mathbf {r} ')}}=\delta (\mathbf {r} -\mathbf {r} ').}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/3eedc9daaf7111863fa1e752f440a8fc86da111b)
离散随机变量的熵是概率质量函数的一个泛函
![{\displaystyle {\begin{aligned}H[p(x)]=-\sum _{x}p(x)\log p(x)\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/96861966ca600ab501ab12eb13a187562581cff1)
于是
![{\displaystyle {\begin{aligned}\left\langle {\frac {\delta H}{\delta p}},\phi \right\rangle &{}=\sum _{x}{\frac {\delta H[p(x)]}{\delta p(x')}}\,\phi (x')\\&{}=\left.{\frac {d}{d\epsilon }}H[p(x)+\epsilon \phi (x)]\right|_{\epsilon =0}\\&{}=-{\frac {d}{d\varepsilon }}\left.\sum _{x}[p(x)+\varepsilon \phi (x)]\log[p(x)+\varepsilon \phi (x)]\right|_{\varepsilon =0}\\&{}=\displaystyle -\sum _{x}[1+\log p(x)]\phi (x)\\&{}=\left\langle -[1+\log p(x)],\phi \right\rangle .\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d1497c79d458d1d7877003999d215ef6d408adc5)
最后,
![{\displaystyle {\frac {\delta H}{\delta p}}=-1-\log p(x).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c0330b79eb37663d7b64c54495b977e1bb4f5dba)
令
![{\displaystyle F[\varphi (x)]=e^{\int \varphi (x)g(x)dx}.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8cf62da868a4878d3d5c56043e0e7947d1a3789f)
以
函数作为测量函数
![{\displaystyle {\begin{aligned}{\frac {\delta F[\varphi (x)]}{\delta \varphi (y)}}&{}=\lim _{\varepsilon \to 0}{\frac {F[\varphi (x)+\varepsilon \delta (x-y)]-F[\varphi (x)]}{\varepsilon }}\\&{}=\lim _{\varepsilon \to 0}{\frac {e^{\int (\varphi (x)+\varepsilon \delta (x-y))g(x)dx}-e^{\int \varphi (x)g(x)dx}}{\varepsilon }}\\&{}=e^{\int \varphi (x)g(x)dx}\lim _{\varepsilon \to 0}{\frac {e^{\varepsilon \int \delta (x-y)g(x)dx}-1}{\varepsilon }}\\&{}=e^{\int \varphi (x)g(x)dx}\lim _{\varepsilon \to 0}{\frac {e^{\varepsilon g(y)}-1}{\varepsilon }}\\&{}=e^{\int \varphi (x)g(x)dx}g(y).\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ec4bb807430d52bf84582815526969c5182132bb)
因此
![{\displaystyle {\frac {\delta F[\varphi (x)]}{\delta \varphi (y)}}=g(y)F[\varphi (x)].}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e78f30af55466f2e117b3dc25af74e86a2db308c)
- ^ 在三維笛卡尔坐标系中,
![{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial \nabla \rho }}={\frac {\partial f}{\partial \rho _{x}}}\mathbf {\hat {i}} +{\frac {\partial f}{\partial \rho _{y}}}\mathbf {\hat {j}} +{\frac {\partial f}{\partial \rho _{z}}}\mathbf {\hat {k}} \,,\qquad &{\text{where}}\ \rho _{x}={\frac {\partial \rho }{\partial x}}\,,\ \rho _{y}={\frac {\partial \rho }{\partial y}}\,,\ \rho _{z}={\frac {\partial \rho }{\partial z}}\,\\&{\text{and}}\ \ \mathbf {\hat {i}} ,\ \mathbf {\hat {j}} ,\ \mathbf {\hat {k}} \ \ {\text{are unit vectors along the x, y, z axes.}}\end{aligned}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/17e7ed61ed5dccd09100ee7792ffe6a997bd2c14)
- ^ 例如,對於三維 (n = 3) 和二階 (i = 2) 導數,張量 ∇(2) 的分量為
![{\displaystyle \left[\nabla ^{(2)}\right]_{\alpha \beta }={\frac {\partial ^{\,2}}{\partial r_{\alpha }\,\partial r_{\beta }}}\qquad \qquad {\text{where}}\quad \alpha ,\beta =1,2,3\,.}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/78cbff2c145d8b1d793a8318a07d6f72883d2f3d)
- ^ 例如,當 n = 3 及 i = 2時,張量的純量積為
![{\displaystyle \nabla ^{(2)}\cdot {\frac {\partial f}{\partial \left(\nabla ^{(2)}\rho \right)}}=\sum _{\alpha ,\beta =1}^{3}\ {\frac {\partial ^{\,2}}{\partial r_{\alpha }\,\partial r_{\beta }}}\ {\frac {\partial f}{\partial \rho _{\alpha \beta }}}\qquad {\text{where}}\ \ \rho _{\alpha \beta }\equiv {\frac {\partial ^{\,2}\rho }{\partial r_{\alpha }\,\partial r_{\beta }}}\ .}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/8614c20390d2fe431b437a27daaae4aa9851b10d)