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Section 18 · Lesson 18.1

Multivariate Calculus

Gradients, Jacobians, and Hessians for functions of many variables.

For a scalar function f:Rn→Rf: \mathbb{R}^n \to \mathbb{R}, the gradient ∇f\nabla f is the vector of partial derivatives:

∇f=(∂f∂x1,…,∂f∂xn)\nabla f = \left(\frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right)

It points in the direction of steepest ascent. The Hessian HH is the matrix of second derivatives:

Hij=∂2f∂xi∂xjH_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}

For vector-valued F:Rn→RmF: \mathbb{R}^n \to \mathbb{R}^m, the Jacobian generalizes the derivative:

Jij=∂Fi∂xjJ_{ij} = \frac{\partial F_i}{\partial x_j}

Critical points are where ∇f=0\nabla f = 0. The Hessian classifies them: positive definite → local min, negative definite → local max, indefinite → saddle. In quant problems, gradients power optimization (mean-variance, model fitting) and the Hessian gives confidence-interval coverage via the inverse Fisher information.