Vector Calculus and PDEs
Conservative
• F conservative if work done to move A to B is independent of the path taken
• vector field: a potential/scal.f 𝜙, on a sim.con Ω, such that 𝐹 = ∇𝜙 in Ω
• force: work performed only depends on force chosen at the start and end points
• Theorem 4.3: following equivalent
1. F a conservative v.f on sim.con domain
2. closed curves 𝐶: ∮𝐶 𝐹 ∙ 𝑑𝒙 = 0
3. for any curves 𝐶1 , 𝐶2 that have the same start/end points: ∮𝐶 𝐹 ∙ 𝑑𝒙 = ∮𝐶 𝐹 ∙ 𝑑𝒙
1 2
• irrotational: if 𝐹 conservative, Stoke’s proves it is also irrotational (cons iff irro)
Convergence
• ̅
Fourier Series: converges at 𝑥 iff its partial sums have a limit: lim 𝑆𝑁 (𝑥) = 𝑓(𝑥)
𝑁→∞
• one-sided limits: 𝑥 approaches c from the right/left gives:
𝑓(𝑐+ ) ≔ lim 𝑓(𝑐 + ℎ) OR 𝑓(𝑐− ) ≔ lim 𝑓(𝑐 − ℎ)
ℎ→0 ℎ→0
ℎ>0 ℎ>0
• Fourier Convergence Theorem: 𝑓 2𝜋 periodic, 𝑓(𝑥) and 𝑓′(𝑥) point.cont on (−𝜋, 𝜋):
∞
1 𝑎0
[ 𝑓(𝑥− ) + 𝑓(𝑥+ )] = + ∑[𝑎𝑛 cos(𝑛𝑥) + 𝑏𝑛 sin(𝑛𝑥)]
2 2
𝑛=1
• extensions: often converge faster than the full series
• Laplace: 2D heat equation converges to the time-independent 2D Laplace equation
Curl
• Ω ∈ 𝑅 3 and 𝑭: Ω → R3 a diff. vector field, the curl of 𝑭 is the vector
𝑒𝑥 𝑒𝑦 𝑒𝑧
𝜕 𝜕 𝜕 𝜕𝐹 𝜕𝐹2 𝜕𝐹 𝜕𝐹1 𝜕𝐹 𝜕𝐹1
∇ × 𝐹 ≔ |𝜕𝑥 𝜕𝑦 𝜕𝑧
| ≔ ( 𝜕𝑦3 − ) 𝑒𝑥 − ( 𝜕𝑥3 − ) 𝑒𝑦 + ( 𝜕𝑥2 − ) 𝑒𝑧
𝜕𝑧 𝜕𝑧 𝜕𝑦
𝐹1 𝐹2 𝐹3
𝜕𝐹
in index form (∇ × 𝐹)𝑖 = 𝜀𝑖𝑗𝑘 𝜕𝑥𝑘
𝑗
• 𝑐𝑢𝑟𝑙 𝐹 for ∇ × 𝐹
• ∇ × (𝜆𝐹 + 𝜇𝐺) = 𝜆∇ × 𝐹 + 𝜇∇ × 𝐺
• ∇ × (𝜙𝐹) = (∇𝜙) × 𝐹 + 𝜙(∇ × 𝐹)
• curl-free/irrotational: if ∇ × 𝐹 = 0 at every point where 𝐹 is defined
• curl acts on a vector and produces a vector
• curl(grad) = 0 = div(curl), curl(curl) is a vector
Dirichlet B.Cs
• the value of the solution specified along the boundary: 𝑢(𝒙) = ℎ(𝒙) 𝑓𝑜𝑟 𝒙 ∈ 𝜕𝐷𝑖
• 1D Heat Eqn, homo: equations and constrains are
𝑥 ∈ [0, 𝐿], 𝑢𝑡 = 𝑘𝑢𝑥𝑥 , 𝑢(0, 𝑡) = 𝑢(𝐿, 𝑡) = 0, 𝑢(𝑥, 0) = 𝑓(𝑥) and the solution is
2 𝜋𝑛 2 𝐿 𝑛𝜋𝑥
𝑢(𝑥, 𝑡) = ∑∞
𝑛=1 𝐵𝑛 𝑒
−𝑘𝜆𝑛 𝑡
sin (𝜆𝑛 𝑥) where 𝜆𝑛 = and 𝐵𝑛 = 𝐿 ∫0 𝑓(𝑥) sin ( ) 𝑑𝑥
𝐿 𝐿
This solution is unique (Thm 18.1)
• (15.3) 1D Heat Eqn, inhomo: equations and constrains are
, 𝑥 ∈ [0, 𝐿], 𝑢𝑡 = 𝑘𝑢𝑥𝑥 , 𝑢(0, 𝑡) = 𝑇0 , 𝑢(𝐿, 𝑡) = 𝑇1 , 𝑢(𝑥, 0) = 𝑓(𝑥), if 𝑇0 = 0 = 𝑇1 then this is the
homo version, to solve the inhomo version propose 𝑢(𝑥, 𝑡) = 𝑈(𝑥) + 𝑢̂(𝑥, 𝑡) then follow
(15.3) to obtain 𝑢̂𝑡 = 𝑘𝑢̂𝑥𝑥 , 𝑢̂(0, 𝑡) = 0, 𝑢̂(𝐿, 𝑡) = 0, 𝑢̂(𝑥, 0) = −𝑈(𝑥) + 𝑓(𝑥)
• 1D Wave Eqn, homo: equations and constrains are
𝑥 ∈ [0, 𝐿], 𝑢𝑡𝑡 = 𝑐 2 𝑢𝑥𝑥 , 𝑢(0, 𝑡) = 𝑢(𝐿, 𝑡) = 0, 𝑢(𝑥, 0) = 𝑢0 (𝑥), 𝑢𝑡 (𝑥, 0) = 𝑣0 (𝑥) and the
𝑛𝜋𝑥 𝑛𝜋𝑐𝑡 𝑛𝜋𝑐𝑡
solution is Thm 16.3 𝑢(𝑥, 𝑡) = ∑∞
𝑛=1 sin ( ) [𝐴𝑛 cos ( ) + 𝐵𝑛 sin ( )] if
𝐿 𝐿 𝐿
2 𝐿 𝑛𝜋𝑥 2 𝐿 𝐿 𝑛𝜋𝑥
𝐴𝑛 = 𝐿 ∫0 𝑢0 (𝑥) sin ( ) 𝑑𝑥 and 𝐵𝑛 = 𝐿 𝑛𝜋𝑐 ∫0 𝑣0 (𝑥) sin ( ) 𝑑𝑥
𝐿 𝐿
• Laplace: 𝑢(𝑥, 𝑦) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ∇ 𝑢 = 0 𝑓𝑜𝑟 (𝑥, 𝑦) ∈ Ω, 𝑢(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) 𝑓𝑜𝑟 (𝑥, 𝑦) ∈ 𝜕Ω
2
This solution if unique (Thm 18.3)
• Laplace, 1 inhomo side: 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 0 𝑓𝑜𝑟 (𝑥, 𝑦) ∈ (0, 𝐿) × (0, 𝐿) with
𝑢(0, 𝑦) = 𝑢(𝐿, 𝑦) = 0 𝑓𝑜𝑟 0 < 𝑦 < 𝐿
{ 𝑢(𝑥, 𝐿) = 0 𝑓𝑜𝑟 0 < 𝑥 < 𝐿 following the 5 steps in 17.2
𝑢(𝑥, 0) = 𝑓(𝑥) 𝑓𝑜𝑟 0 < 𝑥 < 𝐿
• Laplace, 4 inhomo sides: ∇2 𝑢(𝑥, 𝑦) = 0 𝑓𝑜𝑟 (𝑥, 𝑦) ∈ (0, 𝐿) × (0, 𝐿) with
𝑢(𝑥, 0) = 𝑓1 (𝑥), 𝑢(0, 𝑦) = 𝑓2 (𝑦), 𝑢(𝑥, 𝐿) = 𝑓3 (𝑥), 𝑢(𝐿, 𝑦) = 𝑓4 (𝑦) full solutions 17.3
• Laplace eqn in 2D w/ Dir b.c.s: consider ∇2 𝑢 = 0 and 𝑢(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) for (𝑥, 𝑦) ∈ Ω and ∈
𝜕Ω respectively ⟹ Green’s 1st identity ⇒ uniqueness
• Thm 18.3: the soln of Laplace’s eqn in 2D with Dir. b.c.s in unique
• d’Alembert’s Formula: 𝑢𝑡𝑡 = 𝑐 2 𝑢𝑥𝑥 , 𝑢(0, 𝑡) = 𝑢(𝐿, 𝑡) = 0, 𝑢(𝑥, 0) = 𝑢0 (𝑥), 𝑢𝑡 (𝑥, 0) = 𝑣0 (𝑥)
1
let 𝑢
̃0 be the odd 2L-ext, then the solution is: 𝑢(𝑥, 𝑡) = 2 (𝑢
̃(𝑥
0 − 𝑐𝑡) + 𝑢
̃(𝑥
0 + 𝑐𝑡)
Divergence
• Ω ∈ 𝑅 3 and 𝑭: Ω → R3 a cont. diff. vector field, the divergence of 𝐹 is defined by
𝜕𝐹1 𝜕𝐹2 𝜕𝐹3 𝜕𝐹𝑖
• ∇∙𝐹 ≔ + + and in index notation ∇ ∙ 𝐹 =
𝜕𝑥 𝜕𝑦 𝜕𝑧 𝜕𝑥𝑖
• 𝑑𝑖𝑣 𝐹 for ∇ ∙ 𝐹
• divergence is a linear operator
• ∇ ∙ (𝜆𝐹 + 𝜇𝐺) = 𝜆∇ ∙ 𝐹 + 𝜇∇ ∙ 𝐺
• ∇ ∙ (𝜙𝐹) = (∇𝜙) ∙ 𝐹 + 𝜙(∇ ∙ 𝐹)
• divergence-free/solenoidal/incompressible: if ∇ ∙ 𝑢 = 0 at every point where 𝑢 is defined
• acts on a vector and results in a scalar
• div(grad) is a scalar, grad(div) is a vector, div(curl) = 0
Divergence Theorem
• Ω ∈ 𝑅 3 a bounded convex domain with boundary 𝜕Ω =: S and outward pointing unit-
normal vector 𝒏, if 𝐹 is a cont. diff. vector field then:
∭Ω ∇ ∙ 𝑭 𝑑𝑉 = ∬𝑆 𝐹 ∙ 𝑑𝑺 where 𝑑𝑺 ≔ 𝒏 𝑑𝑆 and 𝑆 = 𝜕Ω
• Green’s Theorem: is the divergence thm applied to an area in the plane ℝ2
• Green’s Thm ‘divergence theorem’ form: 𝐷 ∈ ℝ2 a bounded domain with boundary curve
𝜕𝑝 𝜕𝑞 𝑝
̂ 𝑑𝑠
𝐶: ∬𝐷(𝜕𝑥 + 𝜕𝑦) 𝑑𝐴 = ∮𝐶 (𝑞 ) ∙ 𝒏
• both of Green’s Identities follow from the divergence theorem
Fields
• scalar field: a function 𝑓: 𝑅 3 → 𝑅
• vector field: a function 𝐹: 𝑅 3 → 𝑅 3
• component wise: 𝑭 = (𝐹1 , 𝐹2 , 𝐹3 )𝑇 or (𝑭)𝒊 = 𝑭𝒊
