Math 1920 Final formula sheet PDF

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Formula sheet for final exam...

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*add vector symbols to everything Types of integrals: ❑

b

C

a

∫ f (x , y , z)dr=∫ f (r (t))||r ' (t)||dt

Scalar line integral:

, calculate arc length, mass, & electric potential, along

curve C, r(t) for a ≤ t ≤ b ❑

ex: calculate

∫ (x+ y +z )ds

, C is the helix r(t) = (cost, sint, t) and

C

||r ' (t)||=√ ❑

, ds =

||r ' (t)||dt = √ ❑

π

∫ (cost +sint +t) √ ❑ 0 ❑

b

❑

C

a

C

∫ F ⋅ dr=∫ F (r (t))⋅ r ' (t)dt=∫ F 1 dx + F 2 dy + F 3 dz

Vector line integral:

calculate work done and circulation along

a curve C given by r(t) for a ≤ t ≤ b ❑

b

C ❑ ❑

a

∫ F ⋅n ds=∫ F (r (t))⋅ N (t)dt , calculate flux across a curve C given by r(t) for a ≤ t ≤ b

Vector line integral: Surface integral:

❑ ❑

∫❑ ∫ f (x , y , z)dS=∫❑ ∫ f ( G (u , v))||N (u , v )||dudv , S

calculate surface area, total charge, gravitational

D

potential, over a surface with parametrization G(u,v) and parameter domain D ❑❑

❑ ❑

❑ S

❑ ❑

Vector surface integral: ∫ ∫ F ⋅ dS=∫∫ F (G (u , v ))⋅ N (u , v) dudv , calculate flux of a vector field F across surface S with parametrization G(u,v) and parameter domain D How to parametrize various surfaces: Cylinder: G (θ , z)=(Rcosθ , Rsinθ, z) , outward normal = R ¿ , dS=| N||dθdz = Rdθdz 2 Sphere: G (θ , φ)=(Rcosθsin φ , Rsinθsin φ , Rcos φ) , N=R2 sin...