Summary
Vector Calculus Summary Notes
- Module
- Mathematics
- Institution
- University Of Bath (UoB)
A good summary of methods, calculations, theories used in vector calculus
[Show more]
Seller
Follow
Content preview
Vector Calculus and PDEsConservative
• 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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller georgiamersh124. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £2.99. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
76449 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now