Mathematics in the Earth Sciences
Mathematics in the Earth Sciences EART 111
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This 3 page Class Notes was uploaded by Jillian Moore on Monday September 7, 2015. The Class Notes belongs to EART 111 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/182200/eart-111-university-of-california-santa-cruz in Earth Sciences at University of California - Santa Cruz.
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Date Created: 09/07/15
ES 111 Mathematical Methods in the Earth Sciences Equations that you should know Note that this is not a comprehensive list There are things that dont appear on this sheet that I will expect you to know such as how sin7c0s etc are de ned7 or what the differential of cm is You will also need to be able to understand and manipulate these expressions Basic Trigonometry coszx sinzr 1 sinx y sinx cosy cos x siny cosz y coszcosy 7 sinxsiny cosine formula 12 b2 02 7 2bccosA sine formula 1 i b i c sinA T sinB T sinC When ltlt 1 i 962 sinzws and coszmlii Basic Calculus dfii f96hf95 4335 h product rule 7 dU du dx m Tudz Zldz udvuvivdu integration by parts Taylor series expansion 2 3 fa z NO gara f a gm Vectors Vector g a17a27a3 Unit vector gt H m M Vaia a QHQ COS 9 albl 1252 1353 E Dot product Q b Cross product Q X b 0253 a xbz axbi a15370152 a251 Q 9 E 9 sin9 1 Lines and Planes Vector and algebraic equations of a line passing through E zmyo 20 parallel to y 1190 i o y7yoz7zo a b 0 Vector and algebraic equations of a plane passing through L0 07y0720 perpendicular to EWhW 7 70 az7x0by7y0cz7zo0 Partial Differentials if hm f96 M 7 WW dz hHO h The gradient of z ay in two dimensions is given by 5f 5f if 7 7 a The directional derivative Duf of f in the direction Q is given by Duf Q if A critical point occurs when 6f 7 6f 7 0 3m 7 3y 7 For a critical point at a7b D Da7b fwltavbfyyavb 7 fa 11 If D gt 0 and fm or fyy gt0 fab is a minimum If D gt 0 and fm or fyy lt0 fab is a maximum If D lt 0 fab is not a local extremum its a saddle point If D0 indeterminate Vector calculus The del operator 2 731217 Gradient gradf lf 33g Where f is a scalar eld a Divergence div 2 Z y where y 0111203 is a vector eld Curl curl y E y 7 7 7 772 where y 11111203 is a vector eld Matrices Matrices A and the components of a matrix a are always described with row rst7 then column Matrix multiplication Q Ag 71 Cij Z aikbkj k1 where A has 71 columns and E has 71 rows Aj 31 LA in general LBW ETAT where T denotes a transpose matrix Matrices cont7d An inverse matrix A71 is de ned such that A A71 4 1 A L where L is the identity matrix b d 7b i d gt 70 a Here the term ad 7 be is the determinant of A that is lf 4 07 the matrix has no inverse For a 2x2 matrix if A a 71 1 thenA mlt lA El lAll l l5 Al and also lAl lATl Gaussian elimination make sure you know how to do itl Eigenvectorseigenvalues Ag Ag and therefore 4 7 All 0 An n gtlt n matrix always has it different eigenvectors Ordinary Differential Equations If a rst order equation is of the kind a do f 96 dx 9 then it is separable and may be solved by writing f96d96 9ydy If a rst order equation is in standard form 1 19mg W then this equation can be solved by separation after multiplying both sides by the integrating factor 6 1mm Reduction of order if one solution yl to a higher order DE is known7 a second solution may be found by writing yg oy1 and substituting into the original equation Constant eoe eients if presented with a DE of the form any azy a1y aoy 0 where a07 a17 7an are constant7 then there are n independent solutions each of the form y Mm where on is an undetermined constant and inn is a root of the equation anmnl a2m2a1ma0 0 These independent solutions may be combined into a single general solution y elemlm Cgemzm onequotM If you end up with repeated roots7 then the second7 third etc independent solutions are zemm7 zzemwg where m is the root which is repeated 3
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