Taylor Series - Taylor Series in Several Variables

Taylor Series in Several Variables

The Taylor series may also be generalized to functions of more than one variable with

T(x_1,\dots,x_d) = \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \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).\!

For example, for a function that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is:

\begin{align} f(x,y) & \approx f(a,b) +(x-a)\, f_x(a,b) +(y-b)\, f_y(a,b) \\ & {}\quad + \frac{1}{2!}\left, \end{align}

where the subscripts denote the respective partial derivatives.

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

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

where is the gradient of evaluated at and is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes

which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, again in full analogy to the single variable case.

Read more about this topic:  Taylor Series

Famous quotes containing the words taylor, series and/or variables:

    “One sorry fret,
    An anvill Sparke, rose higher,
    And in thy Temple falling, almost set
    The house on fire.
    Such fireballs dropping in the Temple Flame
    Burns up the building: Lord, forbid the same.”
    —Edward Taylor (1645–1729)

    “There is in every either-or a certain naivete which may well befit the evaluator, but ill- becomes the thinker, for whom opposites dissolve in series of transitions.”
    —Robert Musil (1880–1942)

    “The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.”
    —George Steiner (b. 1929)