ES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry and basic calculus


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1 ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems (e.g. mps, orbits), resolving of forces, ngulr vritions (e.g. sttistics), etc. etc. Given the tringle Figure 1 we define sin θ = opp. hyp. = o h Wht do they look like? cos θ = dj. hyp. = h tn θ = sin θ cos θ = opp. dj. = o Note tht cos(θ) = cos( θ) ( even function) nd sin(θ) = sin( θ) ( odd function). In 2D, Pythgors theorem tells us tht h 2 = o Dividing by the squre of the hypotenuse (h 2 ) we get ( ) 2 ( ) o 2 1 = + = cos 2 θ + sin 2 θ h h Note tht cos 2 θ is the sme thing s (cos θ) 2 1
2 The ngle θ is expressed in rdins (which re unitless). When working in rdins, it is simple to clculte the rc length l subtended by the ngle θ: where r is the rdius of the circle. l = rθ Figure 2 Since the perimeter of circle is 2πr, then 360 = 2π rdins. You should lwys use rdins when doing mth problems; degrees re for mps. Useful pproximtions When θ is smll (θ 0) we hve sin θ θ cos θ 1 θ2 2 1 These formule only work if θ is in rdins. We ll see where they come from in Lecture 2. Useful formuls include: 1) Double ngle formule (where does this come from?): sin(θ + φ) = sin θ cos φ + cos θ sin φ cos(θ + φ) = cos θ cos φ sin θ sin φ You cn lso obtin sin nd cos of (θ φ) by replcing φ with φ in the bove formule. You cn lso obtin sin nd cos of 2θ by setting θ = φ. For the following generl tringle we lso hve: 2
3 Figure 3 2) Lw of Cosines 2 = b 2 + c 2 2bc cos α 3) Lw of Sines Exmple pprent dip of fult sin α = b sin β = c sin γ Figure 4 Another exmple: rodcut Differentition When is it useful? Anywhere tht we re interested in chnging quntities (fluid flux, topogrphic grdients, stock mrkets etc. etc.); lso for finding mxim/minim. We will denote ny function of x, for instnce sin(x), x 2, etc., s f(x). It is importnt to understnd tht if f(x) = x 2, then f(x + h) = (x + h) 2. Likewise, if f(x) = cos(x), then f(x + h) = cos(x + h) nd so on. When we tke the slope of line (i.e. its grdient) between two points (x 1, y 1 ) nd (x 2, y 2 ) we clculte it by using y 2 y 1 x 2 x 1 If the y vlues re described by some function of x, y = f(x), then we cn rewrite the slope of the line s Incidentlly, this mkes it obvious tht dx dy dy dx = dx = f(x 2) f(x 1 ) x 2 x 1 = 1/ dy dx. 3
4 Now let s insted write x 2 = x 1 + h. Then s h becomes vnishingly smll, we re clculting the slope of the line t position x 1. The slope of the line described by y = f(x) t position x is known s the derivtive of f(x) with respect to x nd is written dx or f (x). Formlly, we write Wht does this men? Exmple: sin x The bsic rules of differentition re: 1. If f(x) = constnt then dx = 0 2. If f(x) = x n then dx = nxn 1 3. If F (x) = f(x) + g(x) then df dx = dx + dg dx 4. If f(x) = u(x)v(x) then dx 5. If f(x) = g(u(x)) then = dg du dx du dx dx = f f(x + h) f(x) (x) = lim h 0 h dv du = u(x) + v(x) dx dx (chin rule) Specil functions tht you ll need to know re iff(x) = ln(x) then (product rule) dx = x 1 if f(x) = e x then dx = ex if f(x) = sin(x) then dx = cos(x) if f(x) = cos(x) then dx = sin(x) To find the mximum/minimum of function we find the vlue(s) of x t which dx = 0 To determine whether these points re mxim or minim, we differentite second time: if if d 2 f dx 2 = f (x) > 0 d 2 f dx 2 = f (x) < 0 then minimum then mximum Why do things work this wy? Wht hppens if d2 f dx 2 = 0? We ll discuss these issues further in Lecture 8. Exmple: efficient box dimensions 4
5 Integrtion Where is it useful? Grvity, moment of inerti, energy... see exmples below. You cn think of integrtion s the reverse of differentition. Physiclly, differentition involves tking the slope of line, while integrtion involves totlling up the re under curve. Differenttion involves loss of informtion e.g. differentiting f(x) nd f(x)+c will give the sme nswer (f (x)). So when we integrte, we need to dd in n undetermined constnt of integrtion: f (x)dx = f(x) + c where c is constnt. This is n indefinite integrl becuse the limits of integrtion re not defined. If they re defined, then insted we get Simple integrtion exmples b 1. If f(x) = x n then f(x)dx = xn+1 n+1 + c f (x)dx = f(b) f() 2. If f(x) = cos(2x) then f(x)dx = 1 2 sin(2x) + c 3. If f(x) = e x then f(x)dx = 1 ex + c 4. If f(x) = 1 x then f(x)dx = 1 ln x + c Some exmples of when integrtion is useful: 1. Are beneth curve f(x) between x = nd x = b is Are = b f(x)dx Note tht this expression hs the correct units for re. 2. The verge (men) vlue of f(x) between x = nd x = b is Men = 1 b f(x)dx b Agin, note tht this expression hs the correct units. 3. Recll tht work = force times distnce, or dw = f(x)dx where f(x) is the force, dx is the increment in distnce nd dw is the increment in work. The work done in moving from to b is W = b 5 f(x)dx
6 Are the units correct? Exmple: meteor hitting the Erth 4. The lst thing to remember is integrtion by prts. It is esiest to demonstrte how this works with n exmple: xe x dx. To show where this comes from, we strt with the chin rule d(uv) = u dv + v du Integrting both sides yields d(uv) = u dv + v du or u dv = uv v du Alwys choose u so tht if you differentite it enough times it will become constnt. Remember tht you cn lwys check your nswer by differentiting!. Second exmple: e x sin x 6
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