Multivariate Calculus — Section 18: Calculus & Optimization

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

\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 $H$ is the matrix of second derivatives:

H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}

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

J_{ij} = \frac{\partial F_i}{\partial x_j}

Critical points are where $\nabla f = 0$ . 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.

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

\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 $H$ is the matrix of second derivatives:

H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}

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

J_{ij} = \frac{\partial F_i}{\partial x_j}