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# Modeling of Environmental Sciences GEOG 410

UNC

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This 10 page Study Guide was uploaded by Ferne Wiza on Sunday October 25, 2015. The Study Guide belongs to GEOG 410 at University of North Carolina - Chapel Hill taught by Conghe Song in Fall. Since its upload, it has received 43 views. For similar materials see /class/228660/geog-410-university-of-north-carolina-chapel-hill in Geography at University of North Carolina - Chapel Hill.

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Date Created: 10/25/15

1 Quantitative Basis 1 functions polynomial functions exponential functions logarithmic functions trigonometric functions review numerical exercises 2 Derivatives Derivatives of simple functions X 1 Knan1 sinx cosx cosx sinx ex 3 eX ax axlna lnx lx Derivative of compound functions dy 0 du dx Example ysinx2 ux2 then dy dy du 2 7ioi s1nu 39 u 39cos x 2x dx du dx 3 Integration the inverse of derivative Here are a few examples I kdx kx C n1 wax x C n l Ie dx 6x C Icosxdx sinx C Isinxdx cosx C Definitive Integration NewtonLeibniz Formula Ifxdx Fb Fa where fxF x 4 Regression j a bx Sum ofsquares SST SSR SSE R2 Slope intercept Positive correlationnegative correlation II Concepts 1 Ecosystems 2 Components of System Models a reservoirs stocks b processes ows c converters parameters 111 N1 3095 H000 0 d interrelationships 3 feedback loops a positive feedbacks b negative feedbacks 4 System Behavior Patterns Linear Exponential Logistic overshotcrash a Graphic view b System Diagrams c Difference Equations d Impact of model parameters on model behaviors Model calibration Validation and Sensitivity Analysis Three radiation laws a Planck s Equation b StefenBoltzmann s Law c Wien s Law 9939 gt1 Leaf Area Index Ecosystem Productions and Respirations a 9 NPP NEP Ra Rh 9519 0 D ID ID ID ID ID ID I ONUIAUJNt O39 Evapotranspiration Soil water potential Permanent Wilting Point Field Capacity and Soil Porosity Mineralization and immobilization Nitrogen cycle carbon cycle water cycle Greenhouse gases Missing carbon sink Short Questions How do you understand that Earth is a dynamic and evolving system What is the smallest ecosystem you observed in daily life and how life is sustained within the ecosystem What is the solar elevation angle at Chapel Hill latitude36 at noon time on March 21 How does stoma control the loss of water from plants How is transpiration different from evaporation How are soil porosity field capacity and volumetric soil moisture related How relatively humidity vapor pressure saturated vapor pressure vapor pressure deficit are related What are the sources of water for precipitation on land Draw the nitrogen cycling loop for the terrestrial ecosystem Please draw the global watercarbon cycle loop no numbers are needed Essay Questions 1 2 9399 Discuss all possible feedbacks both positive and negative of deforestation on global warming If you would implement the Beer s law as we discussed in class for the entire global what do you need What are the challenges you will face What kind of errors do you expect Discuss how water move in the soil to roots and through plants into the atmosphere How the water potential gradient is created to maintain the ow of water Discuss the factors and the mechanisms that in uence plant photosynthesis Discuss how deforestation can in uence the aquatic animals in the streams that ow out from the area Based on Henry s law can you predict D0 will increase or decrease in a lake for different seasons If yes explain the prediction How does global water cycle in uences global climate Derivatives and Integration 1 Derivatives Example 1 Traveling from Chapel Hill to Raleigh DistanceD30 miles leaving Chapel Hill at 800am arriving Raleigh City Hall at 845am Therefore the speed of your travel is vDt 30miles075hr40 mileshr This actually the average speed you may travel at a different speed at any particular moment If one asks what your speed is at 815am we may have to figure that out in the following way We know Dvt if we know the distance traveled from Chapel Hill at 815am D8 15am and the distance your traveled two minute later ie D815am Then the distance you traveled from 8 15am to 817am is D815amD815am2min then the speed at 815am can be estimated as D815amD817ammile 2 min The actual speed at 8 15 may still be different from the above estimation but it is a better estimation than 40 mileshr In fact the shorter the time you allow your car to travel after 815am the more accurate the speed you calculate Let t stand for time and At for the time allowed for travel the speed at 815am can be written as V N Dt At Dt t78153m At Example 2 How many of you have watched the last launch of space shuttle Endeavour on August 8 2007 How fast the shuttle is traveling at the time it is just off the launch pat How fast the shuttle is traveling at the just before it reached orbit 18000 mileshour In order to get rid of the gravitation of Earth an object has to travel at an accelerating speed of 79kms2 If you do a plot of time and distance the shuttle is traveling it would look like miles seconds The last Endeavour launch took place at 636pm on August 8 If I ask how fast the shuttle is traveling at 637pm how would to gure it out the speed The shorter the At is the more accurate the speed Mathematically V 1 DtAt Dt 2815 A1213 At In general If a function yfX exists at X0 when X increased AX at X0 ie X X0AX the function has a corresponding increase AyfX0AXfxo if the limit of the ratio of Ay to AX eXists when AX9 0 the limit is called the derivative of yfX at XX0 y39 f 39x 1 gzl A13le A1301 Ax Examples yfXC Cconstant This means regardless of what X value is y is always X Thus fXC fXAXC 1 Q1 C C0 AEEIBAx A1301 Ax y Thus the derivative of any constant is zero yxx fXX fXAXXAX y 1 xAx x1 g1 A1301 Ax A1301 Ax A1301 Ax Hm fXX2 fXAXxAX2X22XAXAX2 y 1 Q1 x2 2xAxAx2 x2 1 2xAxAx2 2x A1301 Ax A1301 Ax A1301 Ax In general Kn anl For convenience we can gure out the derivatives for the commonly used functions and put them in a table for later use so that we don t have to do this again and again Here they are C 0 X l Xquot nX 391 sinX c0sX tagc lc0s2X ctagX lsinzx e e ax axlna lnX UK The Geometric Meaning of the Derivatives y x A f 90 Ay 7 ta Ax gap When Ax O the angle p9 0L therefore the derivatives of yfx at x f c is the slope of the tangent line passing x y The functions we provided with derivative are very simple functions We often work with 2 2x more complex functions that are made from the Simple ones for example s1nx e etc we call these functions compound functions as they are functions containing fuctions Where sinxz can be written as sinu where ux2 Similarly e2x can be written as eu where u2x Here are the rules for taking derivatives for compound functions If yfgx is the compound function of yfu and ugx if the derivatives for ugx exists at x and yfu exists at ugx then the derivative of the compound function yfgx with respective to x is dy dy du f f gtxlt a dx du dx or X 11 g X Examples 1 ysinX2 and ysinu uX2 y sinu Xu sinxz XX2 2xcosxz 2 ysin2x Let usinX ysin2Xuza wan dx du dx u2 39 2u2sinx d sinx39 cosx dx 0 2sinx cosx dx du dx 3 yeZXsinx Let u a z du z dx 4 ye2XCOSZX Integration The inverse of derivatives Let me ask the inverse question in Example 1 of derivatives if I travel at 40 mph on I40 east where am I in 45 minutes how far away I am from Chapel Hill We know we traveled 30 miles in 45 min at that speed is that suf cient to know where we are What else do we need to know In derivative we can write dSdt40 The inverse of that is intergration ie j dS 40 j dt S40tC Where C is a constant determined by the initial condition eg X2 2X in fact XZC 2X where C is a constant IZxdx x2 C Similarly we can create a table of integration 1 jkdx kx C n1 2 Ixquotdx x C where nil n l 3 Ie dx e C 4 Icosxdx sinx C 5 Isinxdx cosx C De nite Integration Given a function fX which is bounded on a b Randomly insert n points within a b so that aX0ltX1lt ltXnb separate the interval ab into n smaller intervals X0X1 X1 X2 Kn1 Xn The lengths of the invervals are respectively AX1X1X0 AX2XzX1 AX XnXn1 Take any number s from any interval above calculate the product f8iAXi and sum the product S i f8 Ax Let 7 be the maximum length of the n intervals if X90 regardless of how a b is separated and how si is taken from the interval Xi1 Xi S is always approach a nite limit The limit is the de nite integration of fX in the interval a b Newton Leibniz Formula fxdx Fb Fa where fXF X 17 I f xdx is the area under the curve from a to b Examples 7r2 1 Isinxdx 0 1 2 Iaquot dx 0 3 szdx 2

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