Hessian Matrix | Multivariable Calculus | Khan Academy

Second order partial derivatives are computed by differentiating the function twice. For f(x,y) = x² + y², we get f_xx = 2, f_xy = 0, f_yx = 0, f_yy = 2. For f(x,y,z) = x sin y + y cos z, we compute all nine second order partial derivatives. Clairaut's Theorem states that if second order partial derivatives exist and are continuous, then f_xi_xj = f_xj_xi. The counterexample f(x,y) = (x² - y²)/(x + y) for (x,y) ≠ (0,0) and f(0,0) = 0 shows that without continuity, mixed partial derivatives may differ: f_xy(0,0) = -1 while f_yx(0,0) = 1.

Second order partial derivatives are obtained by differentiating first order partials again: f_xx, f_xy, f_yx, f_yy. Mixed partials f_xy and f_yx involve differentiating in different orders. For f(x,y) = x³y + xy³, f_xy = f_yx = 3x²y + 3xy². The order of differentiation generally matters, but Clairaut's Theorem states that if f and its partial derivatives are continuous in an open region, then mixed partials are equal (f_xy = f_yx). This theorem saves computation time by allowing either order of differentiation. Functions satisfying ∂²f/∂x² + ∂²f/∂y² = 0 are harmonic and solve Laplace's equation, important in heat conduction, fluid flow, and electromagnetics.

To compute second order partial derivatives, first find first order partial derivatives, then differentiate each with respect to the desired variable. For f(x,y) = y·e^(xy) + x³y² - 10, first find f_x = y·e^(xy) + 3x²y², then differentiate again to get f_xx = y²e^(xy) + 6xy². Similarly, find f_y = e^(xy) + 2x³y, then differentiate to get f_yy = xe^(xy) + 2x³. For well-behaved functions, the order of differentiation does not matter: f_xy = f_yx. Clairaut's Theorem states that if f is defined on an open set D in R², and the mixed partial derivatives f_xy and f_yx are continuous throughout D, then f_xy = f_yx at all points in D. Common continuous functions include polynomials, exponential functions, and trigonometric functions.

Second-order partial derivatives involve differentiating twice. There are four types: ∂²f/∂x², ∂²f/∂y², ∂²f/∂x∂y, and ∂²f/∂y∂x. The notation ∂²f/∂x∂y means differentiate with respect to x first, then y. Clairaut's theorem states that if a function is smooth (continuously differentiable), the mixed partial derivatives are equal: ∂²f/∂x∂y = ∂²f/∂y∂x. For f(x,y) = e^(xy), the second partial derivatives are: f_xx = e^(xy)*y², f_yy = e^(xy)*x², and f_xy = f_yx = e^(xy). This equality holds because the function is smooth.

Second partial derivatives are obtained by differentiating first partial derivatives again. Notations include f_xx, f_xy, f_yx, f_yy, or ∂²f/∂x², ∂²f/∂x∂y, etc. For f(x,y) = 3x² + 2xy³ + 3x²y² - 4y, second partial derivatives are: f_xx = 6 + 6y², f_xy = 6y³ + 6x²y, f_yx = 6y³ + 6x²y, and f_yy = 6x²y - 4. Note that f_xy = f_yx. Clairaut's Theorem states that if mixed partial derivatives are continuous on a domain containing a point, they are equal at that point. This means the order of differentiation does not matter for continuous functions. For f(x,y,z) = e^(xy)ln(z), partial derivatives are: ∂f/∂x = y e^(xy) ln(z), ∂f/∂y = x e^(xy) ln(z), and ∂f/∂z = e^(xy)/z.

A linear map f: E → F between vector spaces is completely characterized by its matrix representation. Given a basis u of the domain E and a basis v of the codomain F, the matrix M has m rows (dim F) and n columns (dim E), with the j-th column containing the coordinates of f(u_j) in the basis v. Matrix-vector multiplication Mx computes the coordinates of f(x) in the basis v. The canonical basis in R^n consists of vectors with a single 1 and the rest 0s, and is used by default when no specific bases are mentioned.

For any vector x in V, the representation of T(x) in the basis of W equals the matrix of T multiplied by the representation of x in the basis of V. This means [T(x)]_W = [T]_B × [x]_V, where [T]_B is the matrix representation of T with respect to bases B and W.

Vectors are ordered lists of numbers visualized as arrows in coordinate systems. Matrices are rectangular arrays with rows and columns. Matrix multiplication requires matching inner dimensions (columns of first matrix equal rows of second matrix), with results having outer dimensions. Matrix-vector multiplication transforms vectors through linear operations, where the matrix columns determine how the transformation acts. A linear combination is a sum of scalar multiples of vectors (c1v1 + c2v2 + ... + cnvn). The trivial linear combination (all coefficients zero) always yields the zero vector. A family of vectors is linearly independent if no vector can be expressed as a linear combination of the others. Standard basis vectors (e1, e2, ..., en) have a single 1 and zeros elsewhere. Any vector can be expressed as a linear combination of basis vectors with coefficients equal to its components.

Matrix-vector multiplication requires the matrix to have dimensions m×n and the vector to be n×1 (column vector), producing an m×1 result. The computation is a linear combination of matrix columns weighted by vector elements. This operation can also be viewed as rows of the matrix multiplied by the vector. Matrix-vector multiplication is fundamental for solving linear systems and represents linear transformations in coordinate space.

This section covers the foundational concepts of linear algebra including: (1) Vectors as ordered sets of n numbers, represented as rows (1×n) or columns (n×1); (2) Matrices as rectangular arrangements of m×n numbers; (3) Matrix addition for matrices of the same dimensions, adding corresponding elements; (4) Scalar multiplication multiplying every element by a scalar; (5) Vector-matrix multiplication where a row vector (1×n) times a column vector (n×1) yields a scalar; (6) Matrix multiplication rules requiring columns of the first matrix to equal rows of the second, with non-commutative results.

This segment covers the general Taylor series approximation formula around x = a: Pn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + f^(n)(a)(x-a)^n/n!. The instructor explains that linear approximation corresponds to degree n = 1, while quadratic approximation corresponds to degree n = 2. The instructor emphasizes that for sin(x), the first derivative is cos(x) and the second derivative is -sin(x), which are essential for computing the approximations.

Taylor approximation is a mathematical technique that approximates differentiable functions using polynomials; the first-order Taylor approximation (linear approximation) equals f(x₀) + f'(x₀)(x - x₀), representing the tangent line at the initial point, while the second-order Taylor approximation (quadratic approximation) adds the term (1/2)f''(x₀)(x - x₀)², providing a curved approximation that better captures the function's behavior near the initial point. In economics, these approximations are widely used for analyzing functions around specific points, with examples including e^x ≈ 1 + x and ln(1+x) ≈ x - (1/2)x² when expanded about x=0.

Linear approximation uses the first two terms of Taylor series (f(x₀) + f'(x₀)(x-x₀)) to approximate a function near a point x₀, providing the best linear fit; quadratic approximation extends this by adding the second derivative term (f''(x₀)/2)(x-x₀)², creating a parabolic approximation that better captures curvature. For single-variable functions, common approximations near zero include sin(x) ≈ x, cos(x) ≈ 1, eˣ ≈ 1+x, ln(1+x) ≈ x, and (1+x)ʳ ≈ 1+rx. For multivariable functions, linear approximation becomes f(x₁,x₂,...,xₙ) ≈ f(x₁₀,x₂₀,...,xₙ₀) + Σ(∂f/∂xᵢ|ₓ₀)(xᵢ-xᵢ₀), where each term represents the partial derivative contribution from each variable's deviation from the center point.

Linear approximation uses the first-order Taylor series formula f(x) ≈ f(a) + f'(a)(x-a), while quadratic approximation uses the second-order formula f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)²/2!, both derived from Taylor's Theorem to approximate function values near a point 'a'.

Taylor polynomials approximate general functions with simpler polynomials. The tangent line (degree 1) is given by L(x) = f(a) + f'(a)(x-a), with error satisfying lim_{x→a} R(x)/(x-a) = 0. However, it cannot capture curvature. To improve, we add a quadratic correction α(x-a)², requiring lim_{x→a} R₂(x)/(x-a)² = 0. Using L'Hôpital's rule twice, we derive α = f''(a)/2, yielding P₂(x) = f(a) + f'(a)(x-a) + [f''(a)/2](x-a)².