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Let $\mathbf x=(x_1,x_2,\cdots,x_n), \mathbf a=(a_1,a_2,\cdots,a_n)$, Can we write Taylor's expansion of $f(\mathbf x)$ at $\mathbf a$ as $f(\mathbf a)+(f_1(\mathbf a),f_2(\mathbf a),\cdots,f_n(\mathbf a))(\mathbf x-\mathbf a)^T+\frac{1}{2}(\mathbf x-\mathbf a)\left(\begin{array}{cccc}f_{11}(\mathbf a)&f_{12}(\mathbf a)&\cdots&f_{1n}(\mathbf a)\\f_{21}(\mathbf a)&f_{22}(\mathbf a)&\cdots&f_{2n}(\mathbf a)\\\vdots&\vdots&\vdots&\vdots\\f_{n1}(\mathbf a)&f_{n2}(\mathbf a)&\cdots&f_{nn}(\mathbf a)\end{array}\right)(\mathbf x-\mathbf a)^T+o(\|\mathbf x-\mathbf a\|^2)$ where the norm in the remainder is any norm of $\mathbb R^n$. If it can, please refer me to a textbook that explicitly mentions this. Thank you!

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1 Answer 1

This is multivariable Taylor's formula with the remainder in Peano's form. One should assume that $f$ is twice continuously differentiable. The formula is stated in Encyclopedia of Mathematics where textbook references are given, such as

T.M. Apostol, "Mathematical analysis" , Addison-Wesley (1957)

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