Functions of several variables

This chapter extends differentiation to functions $f:D\to\mathbb{R}$ with $D\subseteq\mathbb{R}^n$, whose inputs can move in many directions. Gradients, tangent planes and local polynomial models describe such functions and turn optimisation, with or without constraints, into systems of equations. The chapter assumes one-variable differentiation and Taylor polynomials.

Lessons

1. Derivatives and level-set geometry: partial derivatives, the gradient as a vector of rates and a normal to level sets and tangent planes.
2. Multivariable approximation: the tangent plane as a first-order Taylor model, the Hessian and Taylor polynomials through total degree three.
3. Local extrema: critical and stationary points and the Hessian test for minima, maxima and saddles.
4. Global extrema: the extreme-value theorem and the interior-and-boundary search on a closed and bounded region.
5. Constrained extrema: Lagrange multipliers for an equality constraint, applied to distance problems.
6. Economic applications: maximising production under a budget, including Cobb-Douglas allocation.
7. Seen questions: past exam questions on partial derivatives of composite functions and the classification of critical points.

Where it leads

The next chapter, double integrals, keeps the same domains, level curves and coordinate geometry but changes the question from local change and optimisation to accumulation over a planar region. The Hessian test links to Algebra through the eigenvalues of a symmetric matrix.

Vocabulary checkpoints

• Gradient and total derivative: the gradient is the vector of partial derivatives; the total derivative is the linear map $\mathbf{h}\mapsto\nabla f\cdot\mathbf{h}$ that it represents at a differentiable point (derivatives and level-set geometry).
• Partial derivatives and differentiability: partial derivatives can exist at a point where $f$ is not differentiable, or not even continuous (local extrema).
• Stationary and critical points: a stationary point has zero gradient; a critical point is an interior point that is either stationary or a point where $f$ is not differentiable (local extrema).
• Local and global extrema: a local extremum compares nearby values; a global extremum on a closed and bounded region also needs every boundary candidate (global extrema).
• Parallel and equal gradients: the Lagrange condition $\nabla f=\lambda\nabla g$ says that the gradients are parallel, not equal, and it is guaranteed only where $\nabla g\ne\mathbf{0}$ (constrained extrema).