References.
OpenStax Calculus Volume 3, Section 4.1
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openstax.org/books/calculus-volume-3/pages/4-1-functions-of-several-variables.
Calculus, Early Transcendentals by Stewart, Section 14.1.
Introduction.
Earth’s surface temperature (\(T\)) at a single time depends on several quantities, the latitude (\(x\)) and longitude (\(y\)) of the point whose temperature is indicated. Mathematically we can describe temperature as a function with two variables or arguments, \(T(x,y)\text{.}\)
Definition 4.1.1.
A real valued function \(f\) of two variables is a rule that, for each ordered pair \((x,y)\) in a subset \(D\) of \(\reals^2\text{,}\) assigns a unique real number denoted by \(f(x,y)\text{.}\) The set \(D\) is the domain of \(f\text{,}\) and the range of \(f\) is the set of all the values that arise for \(f(x,y)\text{.}\)
The rule can be specified with a list of ordered triples \((x,y,z)\) such that in every triple, the initial pair \((x,y)\) is in the set \(D\) (the domain of \(f\)), and for every ordered pair \((x,y)\) in \(D\text{,}\) there is exactly one such triple: the \(z\) value in that triple is denoted \(f(x,y)\text{.}\)
We often write \(z=f(x,y)\text{:}\) \(x\) and \(y\) are the independent variables or arguments of \(f\text{,}\) \(z\) is the dependent variable or value of \(f\text{.}\)
Geometrically, we often think of the domain \(D\) as a set in the plane. When a function is specified by a formulas in terms of the independent variables, the domain is to be all pairs for which the formulas makes sense. This is sometimes called the natural domain associated with the formula. (This can be unclear or hard to determine, so I generally recommend stating the domain explicitly if you can.)
Example 4.1.2.
The (natural) domain of \(f(x,y) = \sqrt{x^2-y^2}\) is all ordered pairs \((x,y)\) with \(|x| \geq |y|\text{.}\) In the plane this is all points in the two quadrants to the right and left of the two lines \(y=x\text{,}\) \(y=-x\text{.}\)
Graphs.
The graph of a function \(f\) is the corresponding set of points \((x,y,z)\) in \(\reals^3\) for which \(z=f(x,y)\text{,}\) so it is typically a surface, like the elliptic paraboloid \(z=x^2+(y/2)^2\text{.}\)
To visualize functons of two variables, I suggest first working out the domain and then examining their graphs with the Desmos 3D Graphing Calculator
3
www.desmos.com/3d
(or other suitable graphing software that you know about).
The Desmos notation for a function of two variables is "\(z = f(x, y)\)" or just the formula "\(f(x, y)\)"; for example, for \(f(x,y) = x^2 + y^2\) in Example 4.2(b) of Section 4.1
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openstax.org/books/calculus-volume-3/pages/4-1-functions-of-several-variables
, one can type in z = x^2 + y^2 (which gets rendered as \(z = x^2 + y^2\)) or just x^2 + y^2.
Level Curves and Contour Plots.
Such surfaces in three dimensions are sometimes hard to illustrate clearly in a two dimensional drawing, so one alternative method of visualization is drawing level curves, also known as contours, as used in maps to show altitude.
Definition 4.1.3.
The level curves of a function \(f\) of two variables are the set of curves in the plane given by the solutions of \(f(x,y)=k\) for each number \(k\) in the range of \(f\text{.}\)
To plot contours at level \(z=k\) with Desmos
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www.desmos.com/3d
the notation is "f(x, y) = k" (or replace the parameter \(k\) by any letter other than the coordinate names \(x\text{,}\)\(y\) or \(z\)). Entering this invites you to add a slider for the parameter \(k\text{.}\) For example, try \(x^2 - y^2 = k\text{.}\)
Example 4.1.4.
For each \(k>0\text{,}\) the level curve of the function given by the formula \(f(x,y)=x^2+(y/2)^2\) is an ellipse.
The level curve for \(k=0\) is"degenerate": it is the single point \((0,0)\text{.}\)
There are no level curves for \(k \lt 0\text{,}\) as such values are not in the range of \(f\text{.}\)
Together the level curves fill out the entire plane.
In general, the collection of all level curves of a function fill out its whole domain, with each point of the domain on exactly one level curve.
Functions of Three or More Variables.
It is easy to extend the above ideas to functions with three or more independent variables, denoted \(f(x,y,z)\) and so on. These are also often natural: for example, air temperature depends on position specified by the three variables latitude, longitude and altitude, and the total energy of a moving object can depend on six variables: three specifying its position and three its velocity. The domain of a function of three variables is a region in three dimensional space \(\reals^3\text{,}\) typically a solid.
Example 4.1.5.
The natural domain of the function given by the formula \(f(x,y,z) = 1/(x^2+y^2+z^2-1)\) is every point in space except those on the sphere of radius 1 center the origin.
Level Surfaces.
Level surfaces are the equivalent here of level curves, and are even more important since the graph of a function of three variables is a four dimensional object, ordered quadruples of numbers \((x,y,z,w)\) with \(w=f(x,y,z)\text{,}\) and these are very difficult to graph or visualize. The level surfaces are sets of points in space, rather easier to draw. Note again that different level surfaces for different \(k\) values do not intersect, and they divide up the domain up into a nested collection of surfaces, like the layers of a deformed onion or a can of pringles.
Example 4.1.6.
The level surfaces of \(f(x,y,z)=x+2y+4z\) are the various planes \(x+2y+4z=k\text{.}\) Note that these are all parallel (they have the same normal \(\vector{1,2,4}\)) and so indeed do not intersect, and together fill out all of \(\reals^3\text{.}\)
Even More Variables, and Vector Valued Arguments.
One quantity can depend on more than three others; say on \(n\) quantities \(x_1 \cdots x_n\text{.}\) We sometimes write such a function with ellipsis as \(f(x_1,x_2,\dots,x_n)\) when \(n\) is large.
At times we consider functions of several variables as a function of a single vector argument:
\(f(x,y,z)\) becomes \(f(\vec{x})\) with \(\vec{x}=\vector{x,y,z}\text{,}\) and
\(f(x,y)\) also becomes \(f(\vec{x})\text{,}\) now with \(\vec{x}=\vector{x,y}\text{.}\)
This is particularly convenient with functions of a large number of variables: defining the \(n\)-component vector \(\vec{x} = \vector{x_1, x_2, \dots , x_n}\text{,}\) the function \(f(x_1, x_2, \dots , x_n)\) can be expressed in the same concise form \(f(\vec{x})\text{.}\)
Study Guide.
Study Section 4.1 of Calculus Volume 3
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openstax.org/books/calculus-volume-3/pages/4-1-functions-of-several-variables
; in particular
All the Definitions.
Examples 1.–6, and the Checkpoints following each.
One or several exercises from each of the following ranges: 5–10, 11–13, 14–29, 30–32.
