Taylor Series in Matrix form

One variable

T(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f^(3)}{3!}(a)(x-a)^3 + ..

or \sum_{n=0}^{\inf} \frac{f^(n)(a)}{n!}(x-a)^n

Several variable

T(x_{1}, \cdots, x_{d}) = \sum_{n_1=0}^{\inf} \cdots \sum_{n_d=0}^{\inf} \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\dots,a_d)

i.e. two variables

f(x,y) = f(a,b) +(x-a)\, f_x(a,b) +(y-b)\, f_y(a,b) + \frac{1}{2!}\left[ (x-a)^2\,f_{xx}(a,b) + 2(x-a)(y-b)\,f_{xy}(a,b) +(y-b)^2\, f_{yy}(a,b) \right] + \cdots

In matrix Form

T(\mathbf{x}) = \sum_{|\alpha| \ge 0}^{}\frac{(\mathbf{x}-\mathbf{a})^{\alpha}}{\alpha !}\,({\mathrm{\partial}^{\alpha}}\,f)(\mathbf{a})

i.e. two variables in matrix form

T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^T\mathrm{D} f(\mathbf{a})  + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^T \,\{\mathrm{D}^2 f(\mathbf{a})\}\,(\mathbf{x} - \mathbf{a}) + \cdots

ref : wiki

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