Week 1 Flashcards
How to construct a unit vector?
v̂ = v/|v| where |v| is magntitude of vector, v
Scalar product between two vectors (in 3 dimensions)
a⋅b = a₁b₁ + a₂b₂ + a₃b₃ = ∑aᵢbᵢ = abcosθ
Cross product between two vectors?
a x b =
|î . ĵ . k̂|
|a₁ a₂ a₃|
|b₁ b₂ b₃|
=abn̂sinθ
= -b x a
Triple product of three vectors?
a ⋅ (b⨯c) =
|a₁ a₂ a₃|
|b₁ b₂ b₃|
|c₁ c₂ c₃|
= c ⋅ (a⨯b)
= b ⋅ (c⨯a)
Vector identity:
a⨯(b⨯c)
a⨯(b⨯c) = (a⋅c)b - (a⋅b)c
Vector identity:
(a⨯b) ⋅ (c⨯d)
(a⨯b) ⋅ (c⨯d) = (a⋅c)(b⋅d) - (a⋅d)(b⋅c)
Coulomb’s law between two point charges?
F = 1/4πε₀ q₁q₂/r²
F₁₂ = q₁₂/4πε₀ r₁-r₂/|r₁-r₂|³
Grad operator?
(cartesian)
Acts on scalar field to return vector field.
∇Φ = ∂Φ/∂x î + ∂Φ/∂y ĵ + ∂Φ/∂z k̂
Div operator?
(cartesian)
Acts on vector field to return scalar field.
∇ ⋅ v = ∂vₓ/∂x + ∂vᵧ/∂y + ∂vᶻ/∂z
∇ ⋅ v > 0 : source
∇ ⋅ v < 0 : sink
Curl operator?
(cartesian)
∇ ⨯ v =
|î . ĵ . k̂|
|∂/∂x ∂/∂y ∂/∂z|
|vₓ vᵧ vᶻ|
Laplacian operator?
(cartesian)
Can act on both vectors and scalars.
∇ ⋅ ∇ = ∇² = ∂/∂x² + ∂/∂y² + ∂/∂z²
Scalar: ∇²Φ = ∂Φ/∂x² + ∂Φ/∂y² + ∂Φ/∂z²
Vector: ∇²v = ∇²vₓ î + ∇²vᵧ ĵ + ∇²vᶻ k̂
3 important vector identities?
curl of grad, div of curl, curl of curl
Curl of Grad is zero
∇ ⨯ ∇Φ = 0
Div of Curl is zero
∇ ⋅ (∇ ⨯ v) = 0
Curl of Curl is Grad of Div - Laplacian
∇ ⨯ (∇ ⨯ v) = ∇(∇ ⋅ v) - ∇²v
What is the divergence theorem?
Links divergence of a volume with flux of field through a surface enclosing that volume.
∫ᵥ∇ ⋅ v dV = ∫ₛv ⋅ dS
where dS = dSn̂, and n̂ is outward facing normal vector of the surface, S.
What is Stokes’ theorem?
Links curl through a surface and the line integral around the edge of that surface.
∫ₛ ∇ ⨯ v dS = ∮ₗ v ⋅ dl
dS = dSn̂, n̂ calculated using right hand rule in relation to line integral direction.
Cartesian to plane polar?
x = rcosθ | y = rsinθ
r̂ = cosθ î + sinθ ĵ
θ̂ = -sinθ î + cosθ ĵ
dxdy = rdrdθ
Cylindrical polar volume element?
dxdydz = rdrdθdz
Spherical polar volume element?
dV = r²sinθ drdθdϕ
Grad/del in cartesian?
∇ψ =
∂ψ/∂x î + ∂ψ/∂y ĵ + ∂ψ/∂z k̂
Div in cartesian?
∇ ⋅ A =
∂Aₓ/∂x + ∂Aᵧ/∂y + ∂Aᶻ/∂z
Curl in cartesian?
∇ ⨯ v =
|î . ĵ . k̂|
|∂/∂x ∂/∂y ∂/∂z|
|vₓ vᵧ vᶻ|
Laplacian in cartesian?
(scalar)
∇²ψ =
∂²ψ/∂x² + ∂²ψ/∂y² + ∂²ψ/∂z²
Grad/del in cylindrical?
∇ψ =
∂ψ/∂r r̂ + 1/r ∂ψ/∂θ θ̂ + ∂ψ/∂z ẑ
Div in cylindrical?
∇ ⋅ A =
1/r ∂/∂r (rAᵣ) + 1/r ∂/∂θ (Aθ) + ∂/∂z (Aᶻ)
Curl in cylindrical?
∇ ⨯ A =
……|r̂ . rθ̂ . ẑ|
1/r|∂/∂r ∂/∂θ ∂/∂z|
……|Aᵣ rAθ Aᶻ|
Laplacian in cylindrical?
(scalar)
∇²ψ =
1/r ∂/∂r (r∂ψ/∂r) + 1/r² ∂²ψ/∂θ² + ∂²ψ/∂z²
Grad/del in spherical?
∇ψ =
∂ψ/∂r r̂ + 1/r ∂ψ/∂θ θ̂ + 1/rsinθ ∂ψ/∂ϕ ϕ̂
Div in spherical?
∇ ⋅ ψ =
1/r² ∂/∂r (r²Aᵣ) +
1/rsinθ ∂/∂θ (Aθsinθ) +
1/rsinθ ∂/∂ϕ (Aϕ)
Curl in spherical?
∇ ⨯ A =
……………|r̂ . rθ̂ . rsinθ ϕ̂|
1/r²sinθ|∂/∂r ∂/∂θ ∂/∂ϕ|
……………|Aᵣ rAθ rsinθAᶻ|
Laplacian in spherical?
(scalar)
∇²ψ =
1/r² ∂/∂r (r² ∂ψ/∂r) + 1/r²sinθ ∂/∂θ (sinθ ∂ψ/∂θ) + 1/r²sinθ ∂²ψ/∂z²
Define the Dirac δ-function
δ(x-a)=0 unless x=a
±∞∫δ(x-a) f(x) dx = f(a)
implying:
±∞∫δ(x-a) dx = 1
δ(x-a) is infinite at a and zero everywhere else. Used to model point distributions.
In 3D: δ⁽³⁾(r-a) = δ(x-a₁)δ(y-a₂)δ(z-a₃)
Normalized Gaussian function
f(x) = 1/√2πσ’ exp(-(x-a)squ / 2σsqu)
Total charge from charge density
Q = ∫ᵥ ρdV
Poisson’s equation
∇²f = -4πg
Laplace’s equation
∇²f = 0
General solution to Poisson’s equation
It can be shown that:
∇(1/|r-r’|) = - r-r’/|r-r’|³
∇²(1/|r-r’|) = -4πδ⁽³⁾(r-r’)
Comparing this to Poisson’s equation, ∇²f = -4πg, allows us to find the general solution.
f(r) = ∫ᵥdV’ g(r’)/|r-r’|
This can be proved by applying the Laplacian to the result.
|r-r’|², where θ’ is the angle between r and r’
|r-r’|²= r² + (r’)² - 2r⋅r’
=r² + (r’)² - 2rr’cosθ’
General solution of Poisson’s equation if only a function of radial coordinate.
Using |r-r’|² = r² + (r’)² - 2rr’cosθ’
f(r) = f(r) = ∞∫ dr’(r’)² ∫ⁿ dθ’ sinθ’ ∫²ⁿ dϕ’ g(r’)/(r²+(r’)² - 2rr’cosθ’)½
ⁿ is pi
