Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question

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)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.