Functions of Several Variables

A function of several variables maps a point in multi-dimensional space to a single real number: $f : \mathbb{R}^n \to \mathbb{R}$, written as $z = f(x, y)$.

Unlike single-variable calculus where a point has only two directions to approach, $\mathbb{R}^2$ allows infinitely many paths, making limits much harder.

The domain is the complete set of input points $(x,y)$ where $f$ is valid.

A level curve is the set of points where $f(x,y) = c$ for a constant $c$. These act like topographic contour lines showing constant values.

Common pitfall: Level curves never cross because each $(x,y)$ has exactly one output value. Densely packed contours indicate steep terrain; confusing contour spacing with function height inverts the map's slope meaning.

Limits and Continuity

The limit $\lim_{(x,y)\to(a,b)} f(x,y) = L$ exists only if $f$ approaches $L$ along every possible path to $(a,b)$.

Testing for non-existence: find two different paths yielding different limits. If they disagree, the limit does not exist.

If any two substitution paths give different values, the multi-variable limit fails entirely.