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Math 211 – Chapter 5 Notes Multivariable Functions and Surfaces
Multivariable Functions Review
We are used to seeing functions of the form y = f ( x ), where x is considered the input variable and
y the output variable. We simply pick an x value, plug it into the function, generate a y value, and the result is a point ( x , y ), which we can then graph on an x - y axis system. If we do this for enough points, we can actually “see” a picture of the function.
Suppose we have a scenario where two input variables are needed to generate the output variable. If we let x and y represent the two inputs, and z the output, we would then have a multivariable function and it would generate points with three coordinates ( x , y , z ). Naturally we want
to plot this point and “see” the graph. But there is a problem. A point with three coordinates needs 3-dimensions to be plotted, and its resulting graph will be a surface instead of a curve.
z = f ( x , y )
Examples of multivariable functions:
Example: Let. We pick two input values, and plug them in. For instance, if
we pick x = 2 and y = 3, we get. The output is z = 10, and we have
generated a point in 3 dimensions called (2,3,10). There are no restrictions on x or y , so we could literally pick every possible pair of inputs for x and y and generate z values. The domain for this function is. If we plotted them all we should get a surface.
z = f ( x , y )= x^2 + 2 y f ( 2 , 3 )=( 2 )^2 + 2 ( 3 )= 10
D = {( x , y }| x , y ∈ℜ }
Example: Let z = (^) xx −+ yy. Just as in the previous example we pick any two values for x and y and
insert them into the function to generate z. However, in this example, we have a denominator to consider. Since the denominator cannot be zero, we need to carefully choose our x and y in such a way that the denominator “ x – y ” does not equal zero. In other words, x cannot equal y. This implies a domain D ={( x , y )| x ≠ y }.
Example: Let (^) z = 1 − x^2 − y^2. We have a radical, and we need to ensure that the expression
inside the radical is not negative, so we set , which after two quick simplification
steps gives us. This is our domain, which we write as. Note
that this is actually a circle on the x - y plane of radius 1, centered at the origin. As long as we pick our x and y values such that they lie inside or on this circle, we won’t run into trouble. In fact, this function is actually the top half of a sphere of radius 1, centered at the origin (0,0,0).
1 − x^2 − y^2 ≥ 0 x^2 + y^2 ≤ 1 D = {( x , y )| x^2 + y^2 ≤ 1 }
Example: The Heat Index I is a temperature figure that is calculated from the actual air temperature T and the relative humidity H. This is a multivariable function I = f ( T , H ). The formula is
complicated so instead the value for I is calculated using a table, which can be found on many websites. For example, a typical June day when the air temperature is T = 110 degrees F and the relative humidity is R = 10% (actually R = 0.1) gives a heat index I = 105 degrees F. In other words, it “feels like” 105 F instead of 110 F. In August when the monsoon is in full swing, we might have a day where the air temperature T = 95 F and the relative humidity R = 60%, and we get a heat index I = 114 F. (Source: http://www.il-st-acad-sci.org/kingdom/geo1013.html )
Graphing and Representing Surfaces.
When we plot a multivariable function z = f ( x , y ), we are attempting to sketch a 3-dimensional
object on a 2-dimension sheet of paper. Unless you are a skilled artist, sketching surfaces can be exceedingly difficult to do by hand. However, there are ways to represent a surface on a 2- dimensional sheet of paper.
Example: Let. We can start generating points but this will get tiresome quickly.
Instead we analyze the function. Note that if we let y = 0 (essentially ignoring the term), we
have , which is a parabola on the xz plane. If we let x = 0, we get another parabola on
the yz plane. The xz and yz planes are perpendicular to one another. We have essentially viewed two cross sections of the surface. We surmise that the surface is a parabolic bowl. Its technical name is paraboloid. A typical satellite dish is a paraboloid, as are your car’s headlight wells (so as to increase reflection and heighten the effect of the light).
z = x^2 + y^2 y^2 z = x^2 z = y^2
Example: Let. Using our trick of setting one variable at a time equal to zero, we get
two “cross sections”: on the xz plane, and
x = 2 x + 3 y z = 2 x z = 3 y on the yz plane. These are both lines, and
we surmise the result is a plane in 3 dimensions. If we face in the direction of positive x , we have a slope of 2, and in the direction of positive y , a slope of 3.
The above “trick” works best for simple functions where the x and y variables are separated out. For more complicated functions we are forced to use computer graphing programs like Maple to generate the graphs.
Example: We’ll re-use our first example. Maple generates this graph. You can see the
parabolic nature of the graph, as though a parabola was turned on its axis to sweep out a 3- dimensional paraboloid.
z = x^2 + y^2
Example: Arizona’s Weather Map:
Air temperature can be thought of as the z variable for a given location, which is denoted by its x and y variables (literally, x is the longitude, y is the latitude).
The map at the left shows contour lines with numbers overlaid on them. These lines are supposedly where the temperature is exactly the given number. Since Phoenix is ‘close’ to the “100” contour line, and since we see that it’s on the plus-side of 100 (do you see why?), we can intrapolate (guess) that Phoenix’s temperature is probably in the low 100s, whereas Yuma’s temperature is in the high 100s. Safford’s is probably in the mid-90s.
The map at right is color coded, and this is how you usually view contour maps in the real world. The information given is exactly the same.
Example: Land form maps (United States Geographical Survey “USGS” Topographical Maps)
Land form maps show the ‘lay of the land’ using contour lines. For example, Humphreys Peak, the highest mountain in Arizona, has a contour map that shows its elevations. Again, think of your x value as your ‘east-west’ position, your y value as your ‘north-south’ position, and the z value will be how high you are in elevation.
Source: www.topozone.com
Note the occasional numbers written in on the thicker contour lines. On this map, the thicker lines differ by 200 feet, with the intervening thinner lines differing by 40 feet. Philomena Spring is approximately 11,580 feet elevation.
