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Calculus Review

Calculus Review. Slope. Slope = rise/run = D y/ D x = (y 2 – y 1 )/(x 2 – x 1 ) Order of points 1 and 2 abitrary, but keeping 1 and 2 together critical Points may lie in any quadrant: slope will work out

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Calculus Review

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  1. Calculus Review

  2. Slope • Slope = rise/run • = Dy/Dx • = (y2 – y1)/(x2 – x1) • Order of points 1 and 2 abitrary, but keeping 1 and 2 together critical • Points may lie in any quadrant: slope will work out • Leibniz notation for derivative based on Dy/Dx; the derivative is written dy/dx

  3. Exponents • x0 = 1

  4. Derivative of a line • y = mx + b • slope m and y axis intercept b • derivative of y = axn + b with respect to x: • dy/dx = a n x(n-1) • Because b is a constant -- think of it as bx0 -- its derivative is 0b-1 = 0 • For a straight line, a = m and n = 1 so • dy/dx = m 1 x(0), or because x0 = 1, • dy/dx = m

  5. Derivative of a polynomial • In differential Calculus, we consider the slopes of curves rather than straight lines • For polynomial y = axn + bxp + cxq + … • derivative with respect to x is • dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  6. Example y = axn + bxp + cxq + … dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

  7. Numerical Derivatives • ‘finite difference’ approximation • slope between points • dy/dx ≈Dy/Dx

  8. Derivative of Sine and Cosine • sin(0) = 0 • period of both sine and cosine is 2p • d(sin(x))/dx = cos(x) • d(cos(x))/dx = -sin(x)

  9. Partial Derivatives • Functions of more than one variable • Example: h(x,y) = x4 + y3 + xy

  10. Partial Derivatives • Partial derivative of h with respect to x at a y location y0 • Notation ∂h/∂x|y=y0 • Treat ys as constants • If these constants stand alone, they drop out of the result • If they are in multiplicative terms involving x, they are retained as constants

  11. Partial Derivatives • Example: • h(x,y) = x4 + y3 + x2y+ xy • ∂h/∂x = 4x3 + 2xy + y • ∂h/∂x|y=y0 = 4x3 + 2xy0+ y0

  12. WHY?

  13. Gradients • del h (or grad h) • Darcy’s Law:

  14. Equipotentials/Velocity Vectors

  15. Capture Zones

  16. Capture Zones

  17. Hydrologic Cycle/Water Balances

  18. Earth’s Water • Covers approximately 75% of the surface • Volcanic emissions http://earthobservatory.nasa.gov/Library/Water/

  19. One estimate of global water distribution http://earthobservatory.nasa.gov/Library/Water/

  20. Fresh Water

  21. Hydrologic Cycle • Powered by energy from the sun • Evaporation 90% of atmospheric water • Transpiration 10% • Evaporation exceeds precipitation over oceans • Precipitation exceeds evaporation over continents • All water stored in atmosphere would cover surface to a depth of 2.5 centimeters • 1 m average annual precipitation http://earthobservatory.nasa.gov/Library/Water/

  22. Hydrologic Cycle In the hydrologic cycle, individual water molecules travel between the oceans, water vapor in the atmosphere, water and ice on the land, and underground water. (Image by Hailey King, NASA GSFC.) http://earthobservatory.nasa.gov/Library/Water/

  23. Water (Mass) Balance • In – Out = Change in Storage • Totally general • Usually for a particular time interval • Many ways to break up components • Different reservoirs can be considered

  24. Water (Mass) Balance • Principal components: • Precipitation • Evaporation • Transpiration • Runoff • P – E – T – Ro = Change in Storage • Units?

  25. Ground Water (Mass) Balance • Principal components: • Recharge • Inflow • Transpiration • Outflow • R + Qin – T – Qout = Change in Storage

  26. Water Balance Components

  27. http://www.srs.fs.usda.gov/gallery/images/5_rain_gauge.jpg

  28. DBHydroRainfall Stations • Approximately 600 stations

  29. Spatial Distribution of Average Rainfall http://sflwww.er.usgs.gov/sfrsf/rooms/hydrology/compete/obspatialmapx.jpg

  30. Voronoi/Thiessen Polygons

  31. Evaporation Pan www.photolib.noaa.gov/ historic/nws/wea01170.htm

  32. Pan Evaporation • Pan Coefficients: 0.58 – 0.78 • Transpiration • Potential Evapotranspiration • Thornwaite Equation

  33. Watersheds http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg

  34. Watersheds http://www.bsatroop257.org/Documents/Summer%20Camp/Topographic%20map%20of%20Bartle.jpg

  35. Stage

  36. Stage Recorder http://gallatin.humboldt.edu/~brad/nws/assets/drum-recorder.jpg

  37. River Hydrograph http://cires.colorado.edu/lewis/epob4030/Figures/UseandProtectionofWaters/figures/ColoradoRiverHydrograph.gif

  38. Well Hydrograph http://wy.water.usgs.gov/news/archives/090100b.htm

  39. Stream Gauging • Measure velocity at 2/10 and 8/10 depth • Q = v*A • Rating curve: • Q vs. Stage http://www.co.jefferson.wa.us/naturalresources/Images/StreamGauging.jpg

  40. http://www.nws.noaa.gov/om/hod/SHManual/SHMan040_rating.htm

  41. Ground Water Basics • Porosity • Head • Hydraulic Conductivity

  42. Porosity Basics • Porosity n (or f) • Volume of pores is also the total volume – the solids volume

  43. Porosity Basics • Can re-write that as: • Then incorporate: • Solid density: rs = Msolids/Vsolids • Bulk density: rb = Msolids/Vtotal • rb/rs = Vsolids/Vtotal

  44. Cubic Packings and Porosity Simple Cubic Body-Centered Cubic Face-Centered Cubic n = 0.48 n = 0. 26 n = 0.26 http://members.tripod.com/~EppE/images.htm

  45. FCC and BCC have same porosity • Bottom line for randomly packed beads: n ≈ 0.4 http://uwp.edu/~li/geol200-01/cryschem/ Smith et al. 1929, PR 34:1271-1274

  46. EffectivePorosity

  47. EffectivePorosity

  48. Porosity Basics • Volumetric water content (q) • Equals porosity for saturated system

  49. Sand and Beads Courtesey C.L. Lin, University of Utah

  50. Aquifer Material (Miami Oolite)

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