Functions of several variables

Functions like temperature $T(x, y, z)$ depend on several variables. Before differentiating them we need:

1. Ways to picture them: graphs and level sets (Def. 1.1).
2. Limits and continuity (Def. 1.2), which are subtler than in one variable because you can approach a point along many paths (Prop. 1.3).
3. Partial derivatives (Def. 1.4): differentiate in one variable, holding the others fixed.

For $f: D\subseteq\mathbb{R}^2\to\mathbb{R}$, the graph is the surface $z = f(x, y)$ in $\mathbb{R}^3$. The level curve (contour) at height $c$ is $\{(x, y): f(x, y) = c\}$. For $f$ of three variables the level sets $f(x, y, z) = c$ are surfaces. Examples: $f = x^2 + y^2$ has circles as level curves (a paraboloid); $f = x^2 - y^2$ has hyperbolas (a saddle); $f = \sqrt{1 - x^2 - y^2}$ is a hemisphere on the disc $x^2 + y^2\le1$.

$\lim_{(x, y)\to(a, b)}f(x, y) = L$ means: for every $\varepsilon > 0$ there is $\delta > 0$ with $|f(x, y) - L| < \varepsilon$ whenever $0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta$. So $f$ must approach $L$ along every path to $(a, b)$. $f$ is continuous at $(a, b)$ if the limit equals $f(a, b)$. Sums, products, quotients (non-zero denominator) and compositions of continuous functions are continuous; polynomials in $x, y$ are continuous everywhere.

If $f(x, y)\to L_1$ along one path to $(a, b)$ and $f\to L_2\ne L_1$ along another, then $\lim_{(x, y)\to(a, b)}f$ does not exist. Agreement along all straight lines is not enough: $f = \frac{x^2y}{x^4 + y^2}$ tends to $0$ along every line through the origin, but to $\frac12$ along $y = x^2$.

the derivative in $x$ with $y$ held fixed; similarly $f_y$. Geometrically $f_x$ is the slope of the graph in the $x$-direction. All the one-variable rules apply, treating the other variables as constants.

Let $f(x, y) = \frac{xy}{x^2 + y^2}$ for $(x, y)\ne(0, 0)$ and $f(0, 0) = 0$. Then $f_x(0, 0) = f_y(0, 0) = 0$ (since $f = 0$ on both axes), yet $f$ is not continuous at the origin (Prop. 1.3 quiz). So in several variables, having partial derivatives is much weaker than being differentiable; the right notion is in Section 2.

1. Continuous at the point? Substitute.
2. Suspect no limit? Try the axes, lines $y = mx$, and curves like $y = x^2$; different answers prove non-existence.
3. Suspect a limit $L$? Bound $|f - L|$ by a function of $r = \sqrt{x^2 + y^2}$ alone (polar coordinates), or squeeze; it must tend to $0$ regardless of $\theta$.
4. Never conclude existence from a few paths.