Lecture 4 on Calculus of Variations You might wonder...If I’m optimizing a shape...a curve, a surface, a whole path, what does "take the derivative and set it to zero" even mean? Do I take the damn derivative with respect to a curve/surface? 🤔 In normal calculus the variable is a number x, so the reflex is clean...f′(x)=0. In calculus of variations the variable is a whole function...the geometry itself, like a curve y(x) (or a surface z(x,y)). So the derivative can’t be a single slope. It has to be a pointwise sensitivity, i.e. how the objective reacts to tiny local deformations. You’re holding a whole shape, like a curve y(x). Your objective isn’t f(x) anymore, it’s a functional J[y], and as we've seen with our first there examples, usually an integral that depends on the entire curve (often through y and y’). To talk about a “derivative”, you do the only thing that makes sense: you nudge the entire curve by a tiny amount and see how J changes. Pick a wiggle shape η(x). It’s not random...it’s any admissible deformation direction. Admissible just means it obeys the constraints. If the endpoints are fixed, you force η(0)=η(1)=0 so the wiggle doesn’t move the endpoints. Then scale that wiggle by a small number ε and define the perturbed curve yε(x)=y(x) εη(x). Now treat ε like the usual scalar in a Taylor expansion. As ε→0, J[y εη] expands as J[y εη] = J[y] ε · (first-order term depending linearly on η) o(ε). So the difference is J[y εη] - J[y] = ε · (linear functional of η) o(ε). For the standard integral of a Lagrangian problems, that linear functional can be written as an inner product with some function of x: J[y εη] - J[y] = ε ∫ (δJ/δy)(x) η(x) dx o(ε). That’s the definition-level meaning of δJ/δy: it’s the unique pointwise sensitivity function that makes this identity true for every admissible η. If δJ/δy is positive at some x, then choosing η negative there decreases J; if δJ/δy is negative there, pushing y upward locally decreases J. It’s literally a map along the curve saying push this way to go downhill. Now translate “set the derivative to zero.” At a minimizer y*, the first-order change must vanish for every admissible wiggle: J[y* εη] − J[y*] = o(ε) for all η. Plug in the expansion and the ε-term must be zero: ∫ (δJ/δy)(x) η(x) dx = 0 for all admissible η. Here’s the crucial logic step: the only way an integral against every test function η can be zero is if the integrand itself is zero (in the usual sense used in analysis). So you get δJ/δy = 0. For the common case J[y]=∫ L(x, y, y’) dx, you can compute δJ/δy explicitly and it becomes the Euler–Lagrange expression δJ/δy = ∂L/∂y − d/dx(∂L/∂y’). So if you name the Euler–Lagrange residual as “left-hand side” R(x) = ∂L/∂y − d/dx(∂L/∂y’), then “set the derivative to zero” is exactly R(x)=0. That’s why animation works so well. You don’t have to solve R=0 in one shot. You can evolve the curve in an artificial time τ by pushing it in the downhill direction: ∂y/∂τ = −R(y). Where the residual is large, the curve moves a lot; as the residual drains toward zero, the motion dies out and the curve settles into an extremal. In our animations, we start from an intentionally ugly curve/surface. Frame by frame the functional drops, the residual drains away, and the geometry relaxes into an extremal. #CalculusOfVariations #EulerLagrange #FunctionalDerivative #GradientFlow #Optimization #MathAnimation

Here is the prompt for this landingpage generated by Gemini 3 Pro @GeminiApp . since its a big prompt expect to try it multiple times to get your best result,cause its always different what an llm gives you as an output (I did polish mine with a second prompt) if yours does not work out, feel free to dm me and i can have a look over it and help you build it right as you want! 🙂 Use it in AI Studio! -------------------------- System Prompt: Senior Creative Frontend Architect Role: You are the Lead Frontend Architect & UI Designer for "Synapse", a Series-B enterprise tech company. Objective: Build the exact website described below. This is an Awwwards "Site of the Day" contender. Standards: Pixel-perfect implementation. 60fps animations. Zero layout shifts. High-fidelity aesthetics. Tech Stack: React 19, Tailwind CSS, GSAP (ScrollTrigger), Three.js (R3F style), Lucide React. --- 1. Design System (The "Organic Enterprise" DNA) Aesthetic: "Stripe meets Sci-Fi". A clean, white-dominant interface with deep "Slate" contrasts and vibrant, electric gradients (Indigo/Violet/Rose) used sparingly. Typography: Plus Jakarta Sans. Headlines: Massive scale (text-7xl to text-9xl), negative tracking (-0.05em), leading-[0.9]. Body: High readability, text-slate-500, leading-relaxed. Code: JetBrains Mono or standard monospace. Surfaces: Light: #ffffff (Surface), #f8f9fa (Canvas). Dark: #0f172a (Slate-900), #1e293b (Slate-800). Glass: backdrop-blur-2xl, bg-white/80, border-white/50. Rounding: Aggressive "Super-Ellipses". Cards use rounded-[2.5rem] or rounded-[3rem]. Shadows: Deep, diffuse shadows: shadow-[0_40px_80px_-20px_rgba(0,0,0,0.1)]. --- 2. Global "Invisible Tech" Layers You must implement these layers to create the "Premium Feel": 1. Noise Overlay: A fixed z-50, pointer-events-none SVG noise filter at 3% opacity. 2. Ambient Light: A radial gradient (rgba(99, 102, 241, 0.08)) tracking the mouse position globally. 3. WebGL Background: A Three.js scene with an infinite grid floor (y=-10) and floating, glass-like icosahedrons that gently orbit. 4. Liquid Filter: A global SVG filter (<filter id="liquid">) for hover distortions. --- 3. Architecture & Routing Do not use React Router. Use a custom state-based view manager in App.tsx. State: currentView: 'home' | 'login' | 'signup' | 'privacy' | 'terms' | 'cookies'. Views: Home: The main scrollable landing page (All sections below). AuthPage: Split-screen Login/Signup. LegalPage: Document layout with sticky sidebar. --- 4. Section-by-Section Implementation Guide (Strict Order) A. Preloader (Cinematic System Boot) Visual: Pristine white. Animation: A complex "Neural Core" SVG logo (concentric rings rotating in opposition). Sequence: Rings assemble -> Lock -> Pulse -> Screen splits into 9 vertical shutters that slide up to reveal the Hero. B. Navigation (Dynamic Island) Behavior: Starts invisible. Slides down after Preloader. Morphing: Starts wide (max-w-[1400px]) and transparent. On scroll, shrinks to a "Pill" (max-w-[800px]), white glass, heavy shadow. C. Hero (The Reveal) Fix: Ensure the final GSAP state sets filter: "blur(0px)" explicitly to prevent rendering fuzziness. Typography: H1: "Intelligence, Synthesized." "Synthesized.": Must use .text-gradient-flow class. IMPORTANT: Apply this class to the individual character spans (via splitText), NOT the parent, to ensure background-clip: text works. Animation: Staggered character reveal (y: 100 -> 0, rotateX: -20deg -> 0). D. Core Capabilities (3D Magnetic Deck) Headline: "Core Capabilities". Sub: "Modular intelligence built for every layer of your enterprise stack." Layout: Floating Masonry. Cards have rounded-[3rem]. Interaction: 3D Magnetic Tilt. Cards rotate (rotateX/Y) based on mouse proximity to cursor. Cards: 1. Neural Processing: "Distributed computation nodes..." (Large Card, Gradient BG). 2. Instant Sync: "Real-time state synchronization..." (Small Card, White). 3. Global Edge: "Content delivery at the speed of light..." (Small Card, White). 4. Adaptive Shield: "AI-driven threat detection..." (Large Card, Gradient BG). Visuals: Giant icons in rounded squares (w-20 h-20). E. Network (Holographic Map) Background: White with precision 1px grid lines that draw themselves (scaleX/Y). Visual: Left: SVG Map with randomized dots 3 pulsing "Hub" nodes connecting arcs. Right: Headline "Global Mesh." (with gradient text). Stats: "PoP Locations" -> "140 " "Capacity" -> "80 Tbps" "Global Latency" -> "< 35ms" "Availability" -> "99.99%" F. Process (Horizontal Scrollytelling) Background: bg-slate-50 (White/Light Grey). NOT DARK. Logic: gsap.to(..., { x: -scrollWidth }) with pin: true. Feel: Heavy friction (scrub: 2). Content: Intro: "System Flow" (Title), "Methodology" (Badge - NO BORDER). Step 1: "01 Ingest" - Real-time topology analysis. Step 2: "02 Deploy" - Edge node provisioning. Step 3: "03 Optimize" - AI-driven route healing. Step 4: "04 Scale" - Infinite horizontal growth. Outro: "Ready to deploy?" (Circular CTA). G. Integration (Built for Developers) Layout: Split Screen. Left (Content): H2: "Built for Developers". Copy: "Our SDKs are typed, documented, and designed to disappear..." List: "Type-safe APIs", "Webhooks & Real-time Events", "Zero-config CLI Tooling", "CI/CD Ready Pipelines". Button: "Read Documentation". Right (Floating Window): Visual: Large Dark Code Window (bg-[#0f172a], rounded-[2rem]). Effect: 3D Tilt on Hover. Code Content (Syntax Highlighting required): import { Synapse } from '@synapse/sdk'; // Initialize client with secure transport const client = new Synapse({ apiKey: process.env.SYNAPSE_KEY, region: 'us-east-1', encryption: true, retryStrategy: Retry.EXPONENTIAL }); // Connect and listen await client.connect(); client.on('stream', (packet) => { console.log('Received:', packet.id); }); UI Element: Blinking _ cursor at the end. H. Architecture (Stacked for Performance) Layout: Left Text, Right 3D Visual. Left: H2: "Stacked for Performance". List: 1. "Edge Logic" - Execute code 5ms from your user. 2. "Distributed State" - Consistency without the latency. 3. "Smart Routing" - Packet optimization at the hardware level. Right (Isometric Stack): Three CSS Layers: "DATA PERSISTENCE" (Bottom, Dark), "COMPUTE MESH" (Middle, Indigo), "APPLICATION" (Top, Glass/White). Animation: Layers separate physically (translateY/Z) as user scrolls. I. DataVis (Real-time Analytics) Visual: Large dark card (bg-slate-900) floating in white section. Headline: "Real-time Analytics". Sub: "Monitor every packet in your network." Stats Bars (Left): "Request Volume" -> 85% (Blue Bar) "Error Rate" -> 0.02% (Green Bar) "Cache Hit Ratio" -> 94% (Purple Bar) Chart (Right): Vertical Bar Chart with 12 bars of varying heights. Hover effect: Tooltip showing "XXk". J. Showcase (Trusted by Industry Leaders) Header: "Trusted by Industry Leaders". Mechanism: Infinite CSS Marquee (@keyframes scroll). Logos: High-quality SVG paths for: "ACME", "Orbit", "Frame", "Vertex", "NEXUS". Effect: Grayscale -> Color on hover. K. Insights (Latest Thinking) Layout: List (Left) vs Visual (Right). List Items: 1. "Engineering" - The Death of Latency. 2. "Product" - Synapse 4.0 Release. 3. "Security" - Zero-Trust Protocol. 4. "Database" - Sharding Logic. Interaction: Hovering a list item updates the abstract shape/color in the Right panel immediately. L. Team (Minds Behind) Layout: Horizontal Accordion. Members: Elena Koch (CTO), Marcus Thorne (COO), Sarah Jenkins (Product), David Chen (Security). Effect: Hover expands card width. Image transitions from Grayscale to Color Liquid Distortion (filter: url(#liquid)). M. Contact (The Void) Design: Massive dark inverted card (bg-slate-900). Headline: "Start Building." Form: Email input with floating label "Create Account" button with slide-fill effect. N. Footer Design: Clean, white. Links: "Privacy Policy", "Terms", "Cookies" must trigger the LegalModal via the View Router. Copyright: "© 2025 Synapse Inc." --- 5. Technical Specifics Tailwind Config: Add animation: { 'spin-slow': 'spin 12s linear infinite' }. Add perspective utilities. GSAP: Use gsap.context() for React safety. CSS: .text-gradient-flow { background-image: radial-gradient(circle at 30% 20%, rgb(99, 102, 241) 0%, transparent 40%), radial-gradient(circle at 80% 40%, rgb(236, 72, 153) 0%, transparent 40%), linear-gradient(135deg, #3b82f6 0%, #8b5cf6 50%, #f97316 100%); background-size: 200% 200%; background-clip: text; -webkit-text-fill-color: transparent; animation: gradientFlow 8s ease infinite alternate; } Output: Generate the complete codebase based on this prompt.

The scientific program at #ICOSAHOM25 starts with a plenary talk by Xiaoying Dai (@CAS__Science) on advances in orthogonality preserving schemes for large scale electron structure calculations, relevant for #quantum #chemistry. Excited to see #gradientflow on the first slide.

Mastering the @Gradient_HQ Building Smarter Neural Networks 🚀 Every neural network learns by adjusting its weights This learning happens through the flow of gradients across the network’s computational graph #GradientFlow #NeuralNetworks #DeepLearning #AIEngineering

論文が出版されました。ピザ屋で雑談しててできた話です DBW2勾配流でinstantonを安定化し、流し続けても位相セクターが跳ばないことを示しました。停止点調整が不要になり系統誤差を抑制できます。BPS解を調べるのにも役立つかも? 詳細→ link.springer.com/article/10… #格子QCD #GradientFlow #Topology

🔓Download Now: #JapaneseJournalofMathematics Published: 06 September 2023 Miroslav Bačák @UniLeipzig: "Old and new challenges in Hadamard spaces" link.springer.com/epdf/10.10… #convexfunction #gradientflow #JJM #100th 🎂 #anniversary

Here's my honest opinion about the GradientFlow app on @webflow youtu.be/JXostNiWVUA Have you used it? What are your thoughts on it?

A big thanks to Ben Lorica 罗瑞卡 at GradientFlow for this recent blog post on weightwatcher. I hope it is useful to you. gradientflow.com/get-the-mos…

This week, added upload capabilities to GradientFlow! Anyone got any embeddings they feel like sharing? Spread the love 😍 loom.com/share/525a8cfd6a7b4… @_buildspace

✨Recently accepted by #informationgeometry, "Conformal mirror descent with logarithmic divergences" by Amanjit Singh Kainth, Ting-Kam Leonard Wong, Frank Rudzicz @UofTStatSci @UofTCompSci #GradientFlow #HalfACenturyOfInformationGeometry 🔓 #OpenAccess link.springer.com/article/10…

GradientFlow作りてえ

Guy Moore @TUDarmstadt gives #SEWM2022 colloquium on transport, the lattice, and noise reduction arXiv:2112.02282 #ShearViscosity #BulkViscosity #NoiseReduction #latticeQCD #GradientFlow

NEW short summary of a recent @GradientFlow newsletter [2021 Silicon Valley Software Engineering Talent Report from @celentialai] ⇢ highlights in the video. You can subscribe at: gradientflow.com/subscribe/

Ever wondered why the winds at your local beach don't match those of your local weather station? It's often due to friction... swellnet.com/news/swellnet-a… #friction #coriolisforce #geostrophicflow #gradientflow #pressuregradientforce #swellnet

🆕 short summary of a recent #GradientFlow Newsletter [Quantitative Finance, #TinyML , Risks in Data & Machine Learning] ⇢ highlights in the video. You can subscribe at: gradientflow.substack.com/su…

🆕 #GradientFlow Newsletter [Day Trading & the Flash Crash, Privacy-preserving analytics, 📈demand for #MachineLearning engineers] ⇢ highlights in the video. You can subscribe at: gradientflow.substack.com/su…

🆕 short summary of a recent #GradientFlow [Modeling Epidemics, AI Infrastructure, and the Power of Business Experiments] ⇢ highlights in the video. You can subscribe at: gradientflow.substack.com/su…

CONGRATULATIONS @GradientFlow! You have won the Go4 charm! Please DM us within 24h to receive it.

機械学習のニュートラルネットワークで、誤差を最小にするためにnegative gradient 方向に動かして、少しずつ臨界点に近づけて行く下りを初めて見たとき、gradientflowでt→∞を日常的にやってた自分には「ひょえ〜」って感じだったよ