Functions of Several Variables: Domains, Level Curves, and Surfaces

A bond option depends on the underlying yield $y$ and time-to-expiry $T$ simultaneously; its price $V(y, T)$ is a function of two variables. A factor-model expected return $\mu(\mathbf{w})$ depends on $n$ portfolio weights at once. Multivariable calculus begins with understanding what these objects look like: their domains (the inputs that make sense), their graphs (surfaces in space), and their level curves (the contours $\{f(x,y)=c\}$ that encode the function's shape without a three-dimensional picture). Level curves are the implicit representation of a surface — they are exactly what the trader sees as a volatility surface "slice" or an option's time-value contour.

Learning objectives. By the end of this note you should be able to: