1 Scalar Function
\(\text{If }f(\mathbf{x})\in\mathbf{R},\mathrm{then}\)
\[df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz=\begin{bmatrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z}\end{bmatrix}\begin{bmatrix}dx\\dy\\dz\end{bmatrix}= f(\mathbf{x})' d\mathbf{x}.
\]
1.1 Derivative
So
\[\dfrac{\partial f}{\partial\mathbf{x}}=\begin{bmatrix}\dfrac{\partial f}{\partial x}&\dfrac{\partial f}{\partial y}&\dfrac{\partial f}{\partial z}\end{bmatrix} = f(\mathbf{x})'
\]
这里用的是 Numerator layout.
1.2 Gradient
\[\left.\nabla f=\left(\frac{\partial f}{\partial\mathbf{x}}\right)^{\mathsf{T}}=\left[\begin{array}{ccc}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z}\end{array}\right.\right]^{\mathsf{T}} = \begin{bmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\\\frac{\partial f}{\partial z}\end{bmatrix}.
\]
2 Vector Function
\(\text{if }\mathbf{f}(\mathbf{x})=\begin{bmatrix}f(\mathbf{x})\\g(\mathbf{x})\\h(\mathbf{x})\end{bmatrix}\in\mathbf{R}^3,\text{then:}\)
2.1 Jacobian
\[\mathbf{J}(\mathbf{x})=\dfrac{\partial\mathbf{f}}{\partial\mathbf{x}}=\begin{bmatrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}&\frac{\partial f}{\partial z}\\\frac{\partial g}{\partial x}&\frac{\partial g}{\partial y}&\frac{\partial g}{\partial z}\\\frac{\partial h}{\partial x}&\frac{\partial h}{\partial y}&\frac{\partial h}{\partial z}\end{bmatrix}
\]
2.2 Divergence
\[\nabla\cdot\mathbf{f}=\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}+\frac{\partial h}{\partial z}
\]
2.3 Curl
\[\nabla\times\mathbf{f}=\begin{bmatrix}\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z}\\\frac{\partial f}{\partial z}-\frac{\partial h}{\partial x}\\\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y}\end{bmatrix}
\]
2.4 Hessian
\[\mathbf{H}=\mathbf{J}(\nabla f(\mathbf{x}))=\begin{bmatrix}\frac{\partial^2f}{\partial x^2}&\frac{\partial^2f}{\partial x\partial y}&\frac{\partial^2f}{\partial x\partial z}\\\frac{\partial^2f}{\partial y\partial x}&\frac{\partial^2f}{\partial y^2}&\frac{\partial^2f}{\partial y\partial z}\\\frac{\partial^2f}{\partial z \partial x}&\frac{\partial^2f}{\partial z \partial y}&\frac{\partial^2f}{\partial z^2}\end{bmatrix}
\]
https://www.value-at-risk.net/functions/