Overview — Multivariable Differential Calculus

The whole of quantitative finance depends on functions that take several inputs and return an output: an option's price depends on the underlying, the strike, time to expiry, and implied volatility simultaneously; a portfolio's variance is a quadratic function of $n$ weights; the Black-Scholes PDE is a partial differential equation in spot and time. This topic builds the mathematical machinery for understanding how such functions behave — when they have limits, when they are smooth, how fast they change in each direction, and what their best local approximation looks like.

By the end of this topic you should be able to:

• describe the domain, range, level curves, and surface of a multivariable function;
• identify and sketch the six standard quadric surfaces (ellipsoid, paraboloid, cone, hyperboloid, saddle);
• evaluate limits along paths and use the two-path test to show non-existence;
• compute partial derivatives $\partial f/\partial x_i$ for scalar-valued functions and state Schwarz's theorem on symmetry of mixed partials;
• distinguish differentiability from mere partial-derivative existence and state the sufficient condition (continuous partials);
• compute the total differential $df = f_x\,dx + f_y\,dy$ and the Jacobian matrix $J = [\partial f_i/\partial x_j]$;
• apply the multivariable chain rule and draw tree diagrams;
• compute the gradient $\nabla f$ and directional derivative $D_\mathbf{u} f = \nabla f \cdot \mathbf{u}$;
• write the equation of the tangent plane to $z = f(x,y)$ at a given point.

1. Contents
2. The big picture