02 Multivariable Functions

Functions, inputs, and outputs

A function of several variables assigns exactly one output to each input from a specified set. For two variables, the standard notation is

Here, $(x,y)$ is the input, $z$ is the output, and $D$ is the domain. A function of three variables may be written $w=f(x,y,z)$. The range is the set of output values obtained from all inputs in the domain.

The variables may represent measurable quantities such as position, time, temperature, pressure, concentration, or elevation. For example, $T(x,y)$ can represent temperature over a flat plate, while $p(x,y,z)$ can represent pressure at a point in space.

To evaluate a function, substitute the input values into its formula. If

then

A formula can impose restrictions on its inputs. For example, $f(x,y)=\sqrt{9-x^2-y^2}$ is real-valued only when $x^2+y^2\le 9$, so its domain is the closed disk of radius $3$ centered at the origin.

Takeaway: Identify the inputs, output, domain, and range before evaluating or interpreting a multivariable function.

Finding domains

The domain consists of every input point for which the formula is defined. For a function of two variables, the domain is a region in the $xy$-plane. Check restrictions systematically:

• A denominator must be nonzero. For $f(x,y)=\frac{1}{x-y}$, require $x-y\ne 0$. The domain is the plane excluding the line $y=x$.

• The radicand of an even root must be nonnegative. For $g(x,y)=\sqrt{4-x^2-y^2}$, require $x^2+y^2\le 4$, which is the disk of radius $2$.

• The argument of a logarithm must be positive. For $h(x,y)=\ln(x+y)$, require $x+y>0$, which describes a half-plane.

When several restrictions occur, impose all of them at once. For

the conditions are

Thus, the domain is the region on or below the line $y=x$, with points on the line $y=-x$ removed.

Takeaway: Translate algebraic restrictions into geometric regions, and use the intersection of all restrictions as the domain.

Graphs and vertical traces

The graph of a function of two variables is the set

It is generally a surface rather than a plane curve. For

the graph is

Because $z\ge 0$, the surface has a lowest point at $(0,0,0)$ and rises as the distance from the origin increases. It is an upward-opening circular paraboloid.

A vertical trace is found by fixing one input variable. Setting $x=0$ gives

which is a parabola in the $yz$-plane. Setting $y=0$ gives

which is a parabola in the $xz$-plane. Other fixed values, such as $x=a$, produce additional cross-sections that help reveal the surface's shape.

Takeaway: Use the graph equation to understand the full surface, and use vertical traces to reduce its analysis to familiar two-dimensional curves.

Level curves and level surfaces

A level curve for a constant $c$ is found by setting

It lies in the $xy$-plane and connects points where the function has the same value. For the paraboloid $f(x,y)=x^2+y^2$, the level curves satisfy

For $c>0$, these are circles centered at the origin with radius $\sqrt{c}$. In particular, $c=1$ gives a circle of radius $1$, $c=4$ gives a circle of radius $2$, and $c=0$ gives only the point $(0,0)$. A collection of level curves forms a contour map. Closely spaced curves indicate relatively rapid change, while widely spaced curves indicate more gradual change.

For the plane $f(x,y)=2x-y$, the level curves satisfy

or equivalently,

They are parallel lines with slope $2$.

For a function of three variables, a level surface is found by setting

For $f(x,y,z)=x^2+y^2+z^2$, the level surfaces are

which are spheres centered at the origin with radius $\sqrt{c}$ when $c>0$.

A level curve of $f(x,y)$ can also be viewed as the intersection of the graph $z=f(x,y)$ with the horizontal plane $z=c$.

Takeaway: Set the function equal to a constant to identify equal-value sets: curves in two input dimensions and surfaces in three input dimensions.

Limits and path testing

The limit

means that function values approach $L$ as the point $(x,y)$ approaches $(a,b)$ from within the domain. The point can be approached along infinitely many paths. The distance to the target point is

so approaching $(a,b)$ means that this distance approaches zero.

Direct substitution often works for expressions assembled from continuous operations when the input is in the valid domain. For example,

To test whether a limit exists, compare paths when the expression is not plainly continuous. Consider

Along $y=0$,

whereas along $x=0$,

Since the paths give different values, the limit at $(0,0)$ does not exist. This is path dependence. Two paths with different results are enough to disprove a limit, but agreement along a few paths is not enough to prove one.

The formal definition states that for every $\varepsilon>0$, there is a $\delta>0$ such that

implies

In words, inputs sufficiently close to $(a,b)$ must produce outputs as close to $L$ as desired.

Takeaway: A multivariable limit must be independent of the path of approach. Use direct substitution when continuity is clear and path comparisons when it is not.

Continuity and removable discontinuities

A function is continuous at a point when its value, nearby limiting behavior, and limiting value agree. At $(a,b)$, verify all three conditions:

1. $f(a,b)$ is defined.

2. $\lim_{(x,y)\to(a,b)}f(x,y)$ exists.

3. $\lim_{(x,y)\to(a,b)}f(x,y)=f(a,b)$.

Polynomials are continuous everywhere. Sums, differences, products, compositions, valid roots, logarithms on their positive-argument domains, trigonometric functions, and rational functions with nonzero denominators are continuous wherever they are defined.

For example,

is continuous everywhere because the polynomial inside the cosine is continuous and cosine is continuous for every real input. The function

is continuous on the region

A removable discontinuity occurs when a limit exists but does not equal the function's assigned value. Consider

For $x\ne y$, factor and simplify:

Therefore,

but $f(1,1)=0$. The function is not continuous at $(1,1)$. Assigning the value $2$ at that point would remove the discontinuity.

Takeaway: Continuity combines domain, limits, and function values. Always check that the point is in the domain before declaring continuity.

A practical analysis workflow

A reliable analysis follows a fixed sequence:

1. Identify the variables and output. Decide whether the function maps from $\mathbb{R}^2$, $\mathbb{R}^3$, or another space into $\mathbb{R}$.

2. Find the domain. Check denominators, even roots, logarithms, and other restrictions.

3. Describe the graph. Use $z=f(x,y)$, recognizable surfaces, and vertical traces.

4. Find level curves or level surfaces by setting the function equal to a constant $c$.

5. Evaluate limits. Try direct substitution when the expression is continuous-looking; otherwise simplify, compare paths, or use bounds or polar coordinates.

6. Check continuity. Confirm that the function is defined, that the limit exists, and that the limit equals the function value.

This workflow connects algebraic formulas with geometric meaning. Domain restrictions determine where the function exists, graphs show its overall shape, level sets display equal values, limits describe local behavior, and continuity determines whether the local behavior matches the function's assigned value.

Final takeaway: For any multivariable function, move from inputs and domain to geometry, then to local behavior through limits and continuity.