Taylor's Formula for Two Variables

Definition

Taylor’s formula extends polynomial approximation to multivariable functions by using higher-order partial derivatives.

Why It Matters

This is the multivariable extension of the ‘best linear approximation.’ It allows engineers to simplify complex 3D surfaces into manageable quadratic models, making optimization of multi-parameter systems (like aerodynamic wings) possible.

Core Concepts

• Linear (first-order) Taylor approximation / tangent plane $L(x, y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$

  • How to read: “The linearization L of x, y equals f of a, b plus the partial derivative f with respect to x at a, b times the quantity x minus a, plus the partial derivative f with respect to y at a, b times the quantity y minus b.”
  • Meaning / when to use: This is the equation of the tangent plane to the surface z = f(x,y) at (a,b). It uses the value at the point plus the two partial derivatives (the slope in x and the slope in y). The best linear guess for f near (a,b). Ubiquitous for linearization in optimization, physics, and error estimation.

• Quadratic (second-order) Taylor approximation $Q(x, y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) + \frac12 \bigl[ f_{xx}(a,b)(x-a)^2 + 2f_{xy}(a,b)(x-a)(y-b) + f_{yy}(a,b)(y-b)^2 \bigr]$

  • How to read: “Q of x, y equals f of a, b plus the partial derivative f with respect to x at a, b times the quantity x minus a, plus the partial derivative f with respect to y at a, b times the quantity y minus b, plus one half times the quantity: the second partial f x x at a, b times x minus a squared, plus two times the mixed partial f x y at a, b times the quantity x minus a times y minus b, plus the second partial f y y at a, b times y minus b squared.”
  • Meaning / when to use: Adds curvature information from the Hessian matrix. The quadratic terms reveal whether the surface curves up, down, or forms a saddle locally—essential for the second-derivative test, Newton’s method in 2D, and understanding local extrema.

• Full Taylor series in two variables (at origin for simplicity) $f(x,y) = f(0,0) + (f_x x + f_y y) + \frac12 (f_{xx}x^2 + 2f_{xy}xy + f_{yy}y^2) + \cdots$

  • How to read: “The function f of x, y equals the function evaluated at the origin, plus the linear terms involving first partial derivatives, plus one half times the quadratic terms involving second partial derivatives, continuing indefinitely.”
  • Meaning: Every sufficiently smooth multivariable function can be approximated near a point by a polynomial whose coefficients are partial derivatives there. The linear term is the gradient dotted with the displacement; the quadratic term is one-half the displacement transposed times the Hessian times the displacement.

• Key mental model: The gradient ∇f gives the direction and magnitude of steepest ascent (first-order change). The Hessian matrix encodes the second-order “bending” — its eigenvalues tell you the principal curvatures. Together they give the local “shape” of the graph up to quadratic order.

Connected Concepts

• Variables (Python)
• Taylor Polynomials
• Functions of Several Variables