 # Function of Two Variables

A function of two variables

The formal definition of a function of two variables is similar to the definition for single variable functions.

A function of two variables f(x, y) has a unique value for f for every element (x, y) in the domain D.

has two inputs (independent variables). A two variable function (bottom) maps a coordinate point (x, y) to a single value (z). The one variable function (top) maps one input (x) to one output (y).

Many common functions have two inputs, including:

• Area of a rectangle with two sides l and w: A = lw
• Work done by force F with displacement d: W = Fd,
• Volume of a right circular cylinder with radius r and height h (V = πr2h). Note that π is a constant and doesn’t count here as a variable.

While a single variable function maps the value of one variable to another, a function of two variables maps ordered pairs (x, y) to another variable.

## Notation

Notation for a function of two variables is very similar to the notation for functions of one variable. For example:

## How to Find the Domain of a Function of Two Variables

The domain is the set of points where the function is defined. Some authors will specify the domain; Many times (especially in real life) you’ll have to figure out what makes sense.

A good starting point is to assume that the domain is all real numbers (from -∞ to ∞), then look for areas where the function doesn’t work. This is where a good knowledge of algebra will come in handy, but if you’re rusty— here are a couple of basic steps (which will catch most of the undefined areas).

Step 1: Ask yourself: Where does this make sense? If a formula is given, like A = lw, then
assume that the domain is defined for points that make sense, unless a specific domain is given. What “makes sense” will depend on the specific situation, but here are a couple of examples:

• Area only makes sense for values of length and width greater than 0.
• Volume only makes sense for positive valued inputs.

Step 2: Look for division by zero.
If a formula has x in a denominator, like this one: Then figure out what would make the denominator zero. The function will be undefined at those points. For this particular formula, the domain is all real numbers (-&infin, ∞) except for x = ±2

Step 2: Look for values that will make quantities under a square root negative. For example:
f(x, y) = √(9 – x2 – y2)
If (x2 – y2) equals anything less than 9, then the value under the square root becomes negative. Therefore, the domain here is all reals except where (x2 – y2) ≤ 9. In more formal notation, you can write that as:
The domain of f(x, y) = {(x, y) ∈ ℝ | x2 – y2 ≤ 9}
Where

• ℝ (doublestruck R) = the set of all real numbers,
• ∈ = “is in the set of”.

## References

Larson, R. & Edwards, B. Multivariable Calculus. Houghton Mifflin.

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