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HARVARD MATH 21A - contour

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Lecture 7: 2/25/2004, FUNCTIONS AND LEVEL CURVES Math21a, O. KnillSection 11.1: 4(abc) 6(abc),14,16,18,42, Section 11.3: 6 18FUNCTIONS, DOMAIN AND RANGE. We deal with functions f(x, y) of two variables defined on a domainD. The domain is usually the entire plane like for f(x, y) = x2+ sin(xy). But there are cases like in f(x, y) =1/p1 − (x2+ y2), where the domain is a subset of the plane. The range is the set of possible values of f.LEVEL CURVES2D: If f(x, y) is a function of two variables, thenf(x, y) = c = const is a curve or a collection ofcurves in the plane. It is called contour curve orlevel curve. For example, f(x, y) = 4x2+ 3y2= 1is an ellipse. Level curves allow to visualize functionsof two variables f(x, y).LEVEL SURFACES. We will later see also 3D ana-logues: if f (x, y, z) is a function of three variablesand c is a constant then f(x, y, z) = c is a surfacein space. It is called a contour surface or a levelsurface. For example if f(x, y, z) = 4x2+ 3y2+ z2then the contour surfaces are ellipsoids.EXAMPLE. Let f (x, y) = x2− y2. The set x2− y2= 0 is the union of the sets x = y and x = −y. The setx2− y2= 1 consists of two hyperbola with with their tips at (−1, 0) and (1, 0). The set x2− y2= −1 consistsof two hyperbola with their tips at (0, ±1).EXAMPLE. Let f(x, y, z) = x2+ y2− z2. f(x, y, z) =0, f(x, y, z) = 1, f(x, y, z) = −1. The set x2+y2−z2= 0 is a conerotational symmetric around the z-axis. The set x2+ y2− z2= 1is a one-sheeted hyperboloid, the set x2+ y2− z2= −1 is atwo-sheeted hyperboloid. (To see that it is two-sheeted notethat the intersection with z = c is empty for −1 ≤ z ≤ 1.)CONTOUR MAP. Drawing several contour curves {f (x, y) = c} or sev-eral contour surfaces {f (x, y, z) = c} produces a contour map.The example shows the graph of the function f(x, y) = sin(xy). We draw the contour map of f: The curvesin(xy) = c is xy = C, where C = arcsin(c) is a constant. The curves y = C/x are hyperbolas except for C = 0,where y = 0 is a line. Also the line x = 0 is a contour curve. The contour map is a family of hyperbolas andthe coordinate axis.TOPOGRAPHY. Topographical maps often show the curves of equal height. With the contour curves asinformation, it is usually already possible to get a good picture of the situation.EXAMPLE. f(x, y) = 1 − 2x2−y2. The contour curves f(x, y) =1 − 2x2+ y2= c are the ellipses2x2+ y2= 1 − c for c < 1.SPECIAL LINES. Level curves are encountered every day:Isobars: pressureIsoclines: directionIsothermes: temperatureIsoheight: heightFor example, the isobars to the right show the lines of constant temper-ature in the north east of the US.A SADDLE. f(x, y) = (x2−y2)e−x2−y2. We can here no morefind explicit formulas for the con-tour curves (x2− y2)e−x2−y2= c.Lets try our best:• f(x, y) = 0 means x2− y2= 0 so that x = y, x = −y are contour curves.• On y = ax the function is g(x) = (1 − a2)x2e−(1+a2)x2.• Because f (x, y) = f(−x, y) = f(x, −y), the function is symmetric with respect to reflections at the x andy axis.A SOMBRERO. The surface z =f(x, y) = sin(px2+ y2) has cir-cles as contour lines.ABOUT CONTINUITY. In reality, one sometimes has to deal with functions which are not smooth or notcontinuous: For example, when plotting the temperature of water in relation to pressure and volume, oneexperiences phase transitions, an other example are water waves breaking in the ocean. Mathematicianshave also tried to explain ”catastrophic” events mathematically with a theory called ”catastrophe theory”.Discontinuous things are useful (for example in switches), or not so useful (for example, if something breaks).DEFINITION. A function f(x, y) is continuous at (a, b) if f(a, b) is finite and lim(x,y)→(a,b)f(x, y) = f(a, b).The later means that that along any curve ~r(t) with r(0) = (a, b), we have limt→0f(~r(t)) = f(a, b).Continuity for functions of more variables is defined in the same way.EXAMPLE. f(x, y) = (xy)/(x2+ y2). Becauselim(x,x)→(0,0)f(x, x) = limx→0x2/(2x2) = 1/2 andlim(x,0)→(0,0)f(0, x) = lim(x,0)→(0,0)0 = 0. Thefunction is not continuous.EXAMPLE. f(x, y) = (x2y)/(x2+ y2). In polarcoordinates this is f(r, θ) = r3cos2(θ) sin(θ)/r2=r cos2(θ) sin(θ). We see that f(r, θ) → 0 uniformlyif r → 0. The function is


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HARVARD MATH 21A - contour

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