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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 don’t appear on this sheet that I will expect you to know (such as how sin,cos etc. are defined, or what the differential of ex^ is). You will also need to be able to understand and manipulate these expressions. Basic Trigonometry
cos^2 x + sin^2 x = 1
sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y − sin x sin y
cosine formula: a^2 = b^2 + c^2 − 2 bc cos A sine formula: a sin A
b sin B
c sin C When x 1: sin x ≈ x and cos x ≈ 1 −
x^2 2 Basic Calculus df dx
= lim h→ 0
f (x + h) − f (x) h product rule: d dx
(uv) = u
dv dx
du dx integration by parts: (^) ∫ udv = uv −
∫ vdu
Taylor series expansion:
f (a + x) = f (a) + xf ′(a) +
x^2 2!
f ′′(a) +
x^3 3!
f ′′′(a) + · · ·
Vectors Vector a = [a 1 , a 2 , a 3 ] Unit vector: aˆ =
a |a|
, |a| =
√ a^21 + a^22 + a^23
Dot product: a · b = |a||b| cos θ = a 1 b 1 + a 2 b 2 + a 3 b 3 Cross product: a ⊗ b = [a 2 b 3 − a 3 b 2 , a 3 b 1 − a 1 b 3 , a 1 b 2 − a 2 b 1 ] |a ⊗ b| = |a||b| sin θ
Lines and Planes Vector and algebraic equations of a line passing through r 0 = (x 0 , y 0 , z 0 ) parallel to v = [a, b, c] :
r = r 0 + tv ,
x − x 0 a
y − y 0 b
z − z 0 c Vector and algebraic equations of a plane passing through r 0 = (x 0 , y 0 , z 0 ) perpendicular to n = [a, b, c] :
n · (r − r 0 ) = 0 , a(x − xo) + b(y − y 0 ) + c(z − z 0 ) = 0 Partial Differentials ∂f ∂x
= lim h→ 0
f (x + h, y) − f (x, y) h The gradient of z = f (x, y) in two dimensions is given by
∇f =
[ ∂f ∂x
∂f ∂y
]
The directional derivative Duf of f in the direction ˆu is given by Duf = ˆu · ∇f A critical point occurs when ∂f ∂x
∂f ∂y
For a critical point at (a, b) D = D(a, b) = fxx(a, b)fyy(a, b) − [fxy(a, b)]^2
If D > 0 and fxx (or fyy) >0: f (a, b) is a minimum If D > 0 and fxx (or fyy) <0: f (a, b) is a maximum If D < 0: f (a, b) is not a local extremum (it’s a saddle point) If D=0: indeterminate Vector calculus The del operator ∇ =
[ (^) ∂ ∂x ,^
∂ ∂y ,^
∂ ∂z
] . Gradient: gradf = ∇f =
[ (^) ∂f ∂x ,^
∂f ∂y ,^
∂f ∂z
] where f is a scalar field. Divergence: div v = ∇ · v = ∂v ∂x^1 + ∂v ∂y^2 + ∂v ∂z^3 where v = [v 1 , v 2 , v 3 ] is a vector field. Curl: curl v = ∇ ⊗ v =
[ (^) ∂v 3 ∂y −^
∂v 2 ∂z ,^
∂v 1 ∂z −^
∂v 3 ∂x ,^
∂v 2 ∂x −^
∂v 1 ∂y
] where v = [v 1 , v 2 , v 3 ] is a vector field. Matrices Matrices A, and the components of a matrix aij , are always described with row first, then column. Matrix multiplication C = AB cij =
∑^ n k=
aikbkj
where A has n columns and B has n rows.
AB 6 = BA (in general)
(AB)T^ = BT^ AT
where T^ denotes a transpose matrix.
If you end up with two complex roots (that is, m 1 = a + ib and m 2 = a − ib) then you can write the solution as y = c 1 eax^ cos bx + c 2 eax^ sin bx Higher Order Partial Differentials If you have a higher-order partial differential equation with a function that depends on two variables (e.g. T (x, t)), try writing T = A(x)B(t) which will give you two separate differential equations, one involving x only and one involving t only. This is called Separation of Variables. The Laplacian of a function f (x, y, · · ·) in Cartesian coordinates is
∇^2 f =
∂^2 f ∂x^2
∂^2 f ∂y^2
