Vectors Flashcards

Circle formula

r² = (x-a)² + (y-b)²

Ellipse formula

1 = x²/a² + y²/b²

Euler’s formula

e^ix = cosx + isinx

Dot product

(a.b) = a_ib_i = |a||b|cosθ

cross product

(a × b)_i = ∈_ijka_jb_k

Scalar Triple product

a . (b x c) = det[a b c]

Vector triple product

a x (b x c)

Gradient

∇ϕ = (∂ϕ​/∂ϕx , ∂ϕ​/∂ϕy , ∂ϕ​/dz)

Divergence

∇⋅F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

Curl

∇ x F = det[ i^ , j^ , k^ ; ∂_x , ∂_y , ∂_z ; F_x , F_y , F_z ]

Laplacian (Scalar)

∇²ϕ = ∂²ϕ​/∂x² + ∂²ϕ​/∂y² + ∂²ϕ/∂z​²

Del dot grad

∇⋅∇ϕ = ∇²ϕ

∇⋅(∇×F) = ?

0

∇×(∇ϕ) = ?

0

∇⋅(ϕF) = ?

ϕ(∇⋅F)+∇ϕ⋅F

∇×(ϕF) = ?

∇ϕ×F+ϕ(∇×F)

All 4 vector identities

∇ ⋅ (∇×F) = 0

∇ × (∇ϕ) = 0

∇ ⋅ (ϕF) = ϕ(∇⋅F) + ∇ϕ⋅F

∇ × (ϕF) = ∇ϕ × F + ϕ(∇×F)

Line integral

∫_C ​F⋅dr

Surface integral

∬_S​ F⋅n dS

Volume integral

∭_V ​f(x,y,z) dV

Divergence theorem

∭_V ​(∇⋅F) dV = ∬_∂V​ F⋅n dS

Stokes’ Theorem

∬_S​(∇×F)⋅n dS = ∮_∂S​ F⋅dr

Material Derivative

D/Dt = ∂/∂t + u . ∇

Continuity equation

∂ρ/∂t + ∇⋅(ρu) = 0

Navier-Stokes (incompressible, steady)

ρ( ∂u/∂t ​+ u⋅∇u ) = −∇p+μ∇²u

Bernoulli’s equation (along streamline, steady)

½​ρu² + p + ρgz = constant

Gauss’s law

∇⋅E = ρ/ε₀​

Gauss for magnetism

∇⋅B = 0

Faraday’s Law

∇ × E= −∂B/∂Bt

Ampere’s Law (with Maxwell correction)

∇ × B = μ₀J + μ₀ε₀ ∂E​/dt

Polar coordinates Grad

∇f = ( ∂f/∂r )𝑒̂_r + [(1/r )( ∂f/∂θ )]𝑒̂_θ

polar coords div

∇⋅F = ( 1/r )[ ∂(rF_r) /∂r + ∂F_θ /∂θ ]

polar coords curl

∇ X F = ( 1/r )[ ∂(rF_θ) /∂r - ∂F_r /∂θ ]

polar coords laplacian

∇²f = ( 1/r )(∂( r∂f /∂r ) /∂r ) + ( 1/r² )(∂²f /∂θ² )

cylindrical coords grad

∇f = ( ∂f/∂R )𝑒̂_R + (1/R ∂f/∂ϕ )𝑒̂_θ + ( ∂f/∂z )𝑒̂_z

cylindrical coords div

∇⋅F = (1/R)[ ∂(RF_R) /∂R + ∂F_ϕ /∂ϕ ] + ∂F_z /∂z

cylindrical coords curl

∇ x F = [(1/R)( ∂F_z /∂ϕ ) - ∂F_ϕ /∂z ] 𝑒̂_R
+ [ ∂F_R /∂z - ∂F_z /∂R ] 𝑒̂_θ
+ [(1/R)( ∂(RF_ϕ) /∂R - ∂F_R /∂ϕ )] 𝑒̂_z

cylindrical coords laplacian

∇²f = (1/R)( ∂( R∂f /∂R ) /∂R )
+ (1/R²)( ∂²f /∂ϕ² )
+ ∂²f /∂z²

Spherical coords grad

∇f = [ ∂f /∂r ] 𝑒̂_r
+ [(1/r)( ∂f /∂θ )] 𝑒̂_θ
+ [( 1/rsinθ )( ∂f /∂ϕ )] 𝑒̂_ϕ

spherical coords div

∇⋅F = [ ∂f /∂r ] 𝑒̂_r
+ [(1/r)( ∂f /∂θ )] 𝑒̂_θ
+ [( 1/rsinθ )( ∂f /∂ϕ )] 𝑒̂_ϕ

spherical coords curl

∇×F =

| 𝑒̂_r r𝑒̂_θ rsinθ𝑒̂_ϕ |
| ∂_r ∂_θ ∂_ϕ | 1 / rsinθ
| F_r rF_θ rsinθF_ϕ |

spherical coords laplacian

∇²f = (1/r² )( ∂ /∂r )( r²∂f /∂r )
+ (1/r²sinθ )( ∂ /∂θ )( sinθ ∂f /dθ )
+ (1/r²sin²θ )( ∂²f /∂ϕ² )