Calculus 3 Functions of Several Variables; Limits and Continuity in Higher Dimensions Question 1: Read the subsection “Graphs, Level Curves, and Contours of Functions of Two Variables” (p. 809). Explain what a level curve is and how it’s related to the surface. The contour curve f(x, y) = 100 - x² - y²-75is the circle x² + y² = 25 in the plane z = 75.z=100-²-²Plane 7510075-The level curve f(x, y) - 100-2²-²-75is the circle x² + y² = 25 in the xy-plane.FIGURE 14.6 A plane z = c paral-lel to the xy-plane intersecting a surfacez = f(x, y) produces a contour curve.The curve in space in which the plane z = c cuts a surface z = f(x, y) is made up ofthe points that represent the function value f(x, y) = c. It is called the contour curvef(x, y) = c to distinguish it from the level curve f(x, y) = c in the domain of f.Figure 14.6 shows the contour curve f(x, y) = 75 on the surface z = 100x² - y²defined by the function f(x, y) = 100x² - y². The contour curve lies directly abovethe circle x² + y² = 25, which is the level curve f(x, y) = 75 in the function's domain.The distinction between level curves and contour curves is often overlooked, and it iscommon to call both types of curves by the same name, relying on context to make it clearwhich type of curve is meant. On most maps, for example, the curves that represent con-stant elevation (height above sea level) are called contours, not level curves (Figure 14.7).Functions of Three VariablesIn the plane, the points where a function of two independent variables has a constant valuef(x, y) = c make a curve in the function's domain. In space, the points where a functionof three independent variables has a constant value f(x, y, z) = c make a surface in thefunction's domain.DEFINITION The set of points (x, y, z) in space where a function of three inde-pendent variables has a constant value f(x, y, z) = c is called a level surface of f. f(x, y) = 7510100The surfacez=f(x, y)= 100-x² - y²is the graph off.f(x, y) = 010f(x,y)=51(a typicallevel curve inthe function'sdomain)FIGURE 14.5 The graph and selectedlevel curves of the function f(x, y) inExample 3. The level curves lie in thexy-plane, which is the domain of thefunction f(x, y).Chapter 14 Partial Derivatives14.1 Functions of Several Variables 809Graphs, Level Curves, and Contours of Functions of Two VariablesThere are two standard ways to picture the values of a function f(x, y). One is to draw andlabel curves in the domain on which f has a constant value. The other is to sketch the sur-face z = f(x, y) in space.DEFINITIONS The set of points in the plane where a function f(x, y) has a con-stant value f(x, y) = c is called a level curve of f. The set of all points (x, y, f(x, y))in space, for (x, y) in the domain of f, is called the graph of f. The graph of f isalso called the surface z = f(x, y).EXAMPLE 3Graph f(x, y) = 100x² - y² and plot the level curves f(x, y) = 0,f(x, y) = 51, and f(x, y) = 75 in the domain off in the plane.Solution The domain of f is the entire xy-plane, and the range of f is the set of realnumbers less than or equal to 100. The graph is the paraboloid z = 100x² - y², thepositive portion of which is shown in Figure 14.5.The level curve f(x, y) = 0 is the set of points in the xy-plane at whichf(x, y) = 100x² - y² = 0, or x² + y² = 100,which is the circle of radius 10 centered at the origin. Similarly, the level curvesf(x, y) = 51 and f(x, y) = 75 (Figure 14.5) are the circlesf(x, y) 100x² - y² = 51,f(x, y) = 100x² - y² = 75,²+²=49x² + y² = 25.orThe level curve f(x, y) = 100 consists of the origin alone. (It is still a level curve.)If x² + y² > 100, then the values of f(x, y) are negative. For example, the circlex² + y² = 144, which is the circle centered at the origin with radius 12, gives the constantvalue f(x, y) = -44 and is a level curve of f.or

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75 The curve in space in which the plane z = c cuts a surface z = f(x, y) is made up of the points that represent the function value f(x, y) = c. It is called the contour curve f(x, y) = c to distinguish it from the level curve f(x, y) = c in the domain of f. Figure 14.6 shows the contour curve f(x, y) = 75 on the surface z = 100-ŕ-y defined by the function f(x, y) = 100-2²-². The contour curve lies directly above the circle x² + y² = 25, which is the level curve f(x, y) = 75 in the function's domain. The distinction between level curves and contour curves is often overlooked, and it is common to call both types of curves by the same name, relying on context to make it clear which type of curve is meant. On most maps, for example, the curves that represent con- stant elevation (height above sea level) are called contours, not level curves (Figure 14.7). Functions of Three Variables

75 The curve in space in which the plane z = c cuts a surface z = f(x, y) is made up of the points that represent the function value f(x, y) = c. It is called the contour curve f(x, y) = c to distinguish it from the level curve f(x, y) = c in the domain of f. Figure 14.6 shows the contour curve f(x, y) = 75 on the surface z = 100-ŕ-y defined by the function f(x, y) = 100-2²-². The contour curve lies directly above the circle x² + y² = 25, which is the level curve f(x, y) = 75 in the function's domain. The distinction between level curves and contour curves is often overlooked, and it is common to call both types of curves by the same name, relying on context to make it clear which type of curve is meant. On most maps, for example, the curves that represent con- stant elevation (height above sea level) are called contours, not level curves (Figure 14.7). Functions of Three Variables

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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