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 z f x y 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 Examples of multivariable functions Example Let z f x y x 2 2 y We pick two input values and plug them in For instance if we pick x 2 and y 3 we get f 2 3 2 2 2 3 10 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 D x y x y If we plotted them all we should get a surface Example Let z x y x y 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 1 x 2 y 2 0 which after two quick simplification steps gives us x 2 y 2 1 This is our domain which we write as D x y x 2 y 2 1 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 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 2dimensional sheet of paper Example Let z x 2 y 2 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 y 2 term we have z x 2 which is a parabola on the xz plane If we let x 0 we get another parabola z y 2 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 Example Let x 2 x 3 y Using our trick of setting one variable at a time equal to zero we get two cross sections z 2 x on the xz plane and 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 z x 2 y 2 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 3dimensional paraboloid Contour Plots A very common way of representing a surface is to use a contour plot In this method curves representing constant values for z are sketched on the surface itself We then orient our viewpoint so that we are looking straight down from the top from the positive z axis The curviness or bumpiness of the graph can t be seen but it can be represented by lines called contours which represent various z values Example Using z x 2 y 2 from our last example we get the following graphs 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 …