Math appendix 1 9/11/03Math appendixImportant numbers, important formulas, important rules, important functions and what they looklike graphically.Numbers worth memorizing:π = 3.14159…e = 2.7183…We often normalize a function or distribution in order to compare it with others. This is shownin many of the graphs below. Normalization is accomplished by dividing both the x-values andthe y-values by constants, and plotting the resulting (now non-dimensional) values on the newaxes. This process is something of an art, which comes in the choice of what scales to use innormalizing. An obvious choice for the constant is the maximum value. Upon dividing by themaximum, the resulting ratios must lie between 0 and 1. Often, however, there is no clearmaximum value (especially on the x-axis), and we must choose something that is characteristicof the problem. Examples include the standard deviation of a distribution, or the period of theoscillation, or the length scale over which the value changes by factor of 2 or of e. (See graphsof exponentials.) The result is always a graph that has values that go from 0 to 1 or from 0 to afew. It also allows us to compare the shapes of functions to one another, as this normalizationremoves the role of the scales themselves. Note for example how all of the Gaussian curves ploton top of one another when normalized.Important functions1. Straight lines. ymxb=+. (Figure 1) Here the slope on the plot is m, while the y-intercept isb. Lines are everywhere in geomorphology. They define the straight slopes of landslide-pronehillslopes. They relate the flux of regolith and slope angle for rainsplash and frost-creepprocesses.2. Negative exponentials (Figure 2) are encountered in radioactive decay, in production profilesof cosmogenic nuclides…yAexx=− /* These are characterized by two constants, A and x*. Thefirst is the maximum value, found at x=0. The second is the scale over which the function fallsby a factor of e. We call this scale the e-folding scale. It is found graphically by finding theplace at which the value of the function is 1/e of A, or A/e. Recalling that e is about 3 (actually2.7183…), this is roughly a third of the value of A, which is easy to estimate on the graph.These functions are encountered in the decay of radioactive nuclides.3. Positive exponentials (Figure 3) are found in the unchecked growth of populations:yAexx=/*. In this case, x would stand for time. Exponential growth is what we expect in apopulation that grows at a rate dictated by the number of individuals in the population at anytime. Just as negative exponentials, it too is characterized by two constants, A and x*. The firstis again the initial value, found at x=0. The second is the scale over which the function changes(this time increases) by a factor of e (the e-folding scale).Math appendix 2 9/11/034. Closely related to exponentials is a function that approaches an asymptote as an exponential(Figure 4). yA exx=−−()/*1. Here the function is defined by the value of the asymptote, A, andby the rate at which the asymptote is approached. Again, this is set by a scale, we use x*. Notethe values of the function at three places: at x=0, e(0) = 1, implying y = 0. At x =∞, e()−∞= 0,implying y = A. Finally, at x=x*, y=A(1-e-1) = A(1-(1/e)) = 0.63A. This approach toward anasymptote is found in systems in which both growth and decay occur, the asymptote reflecting abalance of growth and decay (also called secular equilibrium in radioactive decay series).5. Power law functions (Figure 5) are very common in geomorphology. yAxp= They arise indrainage basin characteristics… They have the important property that they become straightlines when plotted on log-log graphs. The slope of the line is the power. You can see this bylogging both sides of the equation: log( ) log( ) log( )yApx=+. This has the form of ybmx=+which we all recognize as a straight line with intercept b and slope m. This means that an easyway to evaluate the power in a power-law function is by plotting it in this manner. Power lawfunctions are everywhere in geomorphology. To name two examples, power laws describe therelationship between the number of streams of one order with respect to the number in the nextorder (one of Horton’s laws), and the relationship between slope and drainage area in a bedrockstream profile.6. The parabola (Figure 6) is of course a special case of a power law. yy Axxoo=+ −()2. Butit is so commonly seen in geomorphology it is worth breaking out separately. Sand grainssplashed up by raindrops carry out parabolic trajectories. Steady state hilltops have parabolictopographic profiles. Flow of a viscous fluid between two plates (like magma in a dike) has aparabolic velocity profile. The general form here accommodates a parabola centered not on [0,0]but on [xo,yo]. This can be seen by setting x = xo. The value of y = yo.7. Logarithmic functions (Figure 7). y=A log(x/xo). These are encountered in fluid mechanics.For example, the flow speed increases as a logarithm of height above the bed in an open channelflow. These functions have the property of rapidly increasing at first and much more slowlyincreasing thereafter.8. Plots of two major trigonometric functions (Figure 8) are shown in order to emphasize therole of scaling in both the x and y dimensions. Recall that sin(0) = 0 and that cos(0) = 1. Quitecomplicated looking graphs can be constructed by combining sin and cosine curves – this is theessence of the Fourier transform. The surface temperature of the earth carries out sinusoidalswings on both daily and annual time scales.9. The hyperbolic tangent (Figure 9). We include this function because it serves to stepsmoothly from one value (here b-a) to another (b+a) over a specified distance, scaled by x* andcentered at xo. The formula is y = b + a tanh((x-xo)/x*).10. Hyperbolic sine and hyperbolic cosine (Figure 10) can also be scaled as shown on theirplots. One encounters these functions in solutions for the displacement profile around a fault.11. The Gaussian (Figure 11) is often encountered in error analysis, as errors are supposed to benormally, or Gaussianly distributed. The function is named for Karl Freidrich Gauss, a greatMath appendix 3 9/11/03German mathematician and astronomer living 1777-1855. yAexxxo=−−(*)2This is the