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Math Analysis 2

Functions of several variables: domain, level curves, limits and continuity

Topology of R^n, graphs and level curves, limits with examples of non-existence along lines and parabolas, polar coordinates, continuity and Weierstrass' theorem.

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The temperature in a room depends on three coordinates, the resistance of a thermistor on temperature and current, the gain of an amplifier on frequency and load. Almost every real quantity depends on several variables. In this lesson we extend the ideas of Analysis 1 to functions f: \mathbb R^n\to\mathbb R: how they are represented, what neighbourhoods, open and closed sets are, and how limits and continuity are defined.

The space \mathbb R^n

The points of \mathbb R^n are n-tuples \mathbf x = (x_1, \dots, x_n). Distance is Euclidean, \|\mathbf x - \mathbf y\| = \sqrt{\sum(x_i - y_i)^2}. The ball (or neighbourhood) with centre \mathbf x_0 and radius r is B_r(\mathbf x_0) = \{\mathbf x : \|\mathbf x - \mathbf x_0\| \lt r\}: an interval in \mathbb R, a disc in the plane, a solid sphere in space.

Relative to a set A, a point can be:

  • interior: there is a ball around it entirely inside A;
  • boundary: every ball around it contains points of A and points outside A;
  • accumulation point: every ball around it contains points of A other than itself.

A is open if all its points are interior (the disc without its edge), closed if it contains its whole boundary (the disc with its edge), bounded if it fits inside a ball. A closed and bounded set is called compact: it is the counterpart of the intervals [a, b] of Analysis 1, and Weierstrass’ theorem still holds on it.

Representing a function of two variables

The graph of f(x, y) is the surface z = f(x, y) in space. Since surfaces are awkward to draw, level curves are used: the sets \{(x, y) : f(x, y) = c\}, projected onto the xy plane. It is exactly the method of topographic maps (contour lines), weather maps (isobars) and electric field maps (equipotential lines).

Graph of a function of two variables with level curves on the surface and on the plane
The hill-shaped surface z = f(x, y) and some of its level curves: cutting the surface at constant heights gives curves which, projected onto the xy plane, form the function’s “contour map”.

Where the level curves (at equally spaced heights) are close together the function rises quickly; where they are far apart the surface is almost flat.

Examples. f(x, y) = x^2 + y^2 has concentric circles as level curves and a paraboloid as graph. f(x, y) = x^2 - y^2 has hyperbolas as level curves and a “saddle” as graph. f(x, y) = ax + by has parallel lines and a plane as graph.

Domain. As in Analysis 1 you need to find where the expression makes sense, but now the result is a region of the plane. For f(x, y) = \sqrt{4 - x^2 - y^2} + \ln(y - x) you need x^2 + y^2 \le 4 and y \gt x: the disc of radius 2 intersected with the half-plane above the diagonal.

Limits

    \[ \lim_{\mathbf x\to\mathbf x_0} f(\mathbf x) = L \]

if for every \varepsilon \gt 0 there is \delta \gt 0 such that |f(\mathbf x) - L| \lt \varepsilon for all \mathbf x in the domain with 0 \lt \|\mathbf x - \mathbf x_0\| \lt \delta. The definition is identical to Analysis 1, but with a huge difference: in the plane you can approach the point from infinitely many directions and along infinitely many paths, and the limit must be the same for all of them.

How to show a limit does not exist

It is enough to find two paths along which the limits differ. Example:

    \[ f(x, y) = \frac{xy}{x^2 + y^2},\qquad (x, y)\to(0, 0) \]

Along the x axis (y = 0) it is always 0. Along the line y = mx it equals \frac{m x^2}{(1 + m^2)x^2} = \frac{m}{1 + m^2}, which depends on m: along the diagonal it is 1/2. The limit does not exist.

Careful: having the same limit along every line is not enough. For g(x, y) = \frac{x^2 y}{x^4 + y^2} the limit along every line through the origin is 0, but along the parabola y = x^2 the function is constantly 1/2.

How to show a limit exists: polar coordinates

Set x = \rho\cos\theta, y = \rho\sin\theta, and look for a bound |f - L| \le g(\rho) with g(\rho)\to 0, independent of \theta. Example:

    \[ \left|\frac{x^2 y}{x^2 + y^2}\right| = \frac{\rho^3|\cos^2\theta\sin\theta|}{\rho^2} \le \rho \to 0 \]

so the limit is 0. It is not enough that for each fixed \theta the limit in \rho is zero: the bound must be uniform in \theta, otherwise you fall back into the parabola case.

Continuity

f is continuous at \mathbf x_0 if \lim_{\mathbf x\to\mathbf x_0}f(\mathbf x) = f(\mathbf x_0). Sums, products, quotients (where the denominator is non-zero) and compositions of continuous functions are continuous: polynomials in x and y, e^{xy}, \sin(x^2 + y) are continuous everywhere.

The function \frac{xy}{x^2 + y^2}, extended by 0 at the origin, is not continuous at (0, 0), even though it is continuous along every horizontal and vertical line: the first sign that things are more delicate in several variables.

Weierstrass’ theorem. A continuous function on a compact set has a maximum and a minimum. We will use it to look for absolute extrema in the lessons on maxima and minima.

Vector-valued functions

More generally one studies functions \mathbf F: \mathbb R^n\to\mathbb R^m. With n = 1 they are curves (the position of a point as a function of time); with n = m = 2 or 3 they are vector fields, such as the electric field or the velocity of a fluid, which attach a vector to every point. Limits and continuity are studied component by component.

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