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Unformatted text preview: LESSON 7 Functions of Several Variable and Their Limits READ: Sections 15.1, 15.2 NOTES: There are many situations in which a quantity depends on several other quantities. For example, the volume V of a circular cylinder depends on both the height, h , of the cylinder and on the radius, r , of the base: V = πr 2 h . In such a case, we say that V is a function of two variables. In general, if z depends in some way on the quantities x and y , we’ll say z is a function of two variables and write z = f ( x,y ). As pointed out a few lessons back, it is possible to draw the graph of an equation z = f ( x,y ) in 3-space. Recall that as a rough general guide, the graph will look like a sheet floating above (or below) a region in the x , y-plane. The graphs of a few such surfaces are already familiar: paraboloids, hyperboloids, and so on. One way to get an idea about the shape of the graph of a less familiar surface, z = f ( x,y ), is to look at the traces of the surface on planes parallel to the x , y-plane. In other words, look at the curves produced by intersecting the surface with planes of the form z = k for a number of constants k . If we think of the surface as a mountain, then the trace would be the all the points on the mountain at an elevation of k . Such curves are called contour curves or level curves . If these curves are projected down onto the x , y-plane, we get a 2-dimensional representation of the surface. When the TV weather forecaster draws in the isobars on the weather map, it is actually level curves that are being drawn. There is a function of two variables in the background: z = p ( x,y ) gives the pressure at the point ( x,y ) on the earth (which we can pretend is a plane for this example). Isobar means equal pressure , and each isobar represent a line along which the pressure has some fixed value. It is easy to find the equations for the level curves. Simply set z = k , a constant. The result will be an equation in just the two variables x and y which we can graph in the x , y-plane. As an easy example, for z = x 2 + y 2 , when z is set equal to k , the equation for the level curve is seen to be x 2 + y 2 = k . So the level curves are circles centered at the origin in the x , y-plane, at least when k is a positive value. If level curves are drawn for several values of z = k , we can begin the get a feeling for the general shape of the surface. More about the nature of the surface can be deduced from the level curves if they are plotted for a number of equally spaces values of k . If you think about the mountain analogy, you will see that when such level curves are close together, the mountain is very steep, while widely separated levels curves indicate a flat spot on the mountain. Or, back to the weather map analogy, when the level curves are close together, it means that nearby places have a large difference in pressure, and hence it will be pretty windy there as air rushes from the higher pressure area to the lower pressure area. On the other hand, widely separatedair rushes from the higher pressure area to the lower pressure area....
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