Stokes’ Theorem Made Simple! Converting Line Integrals to Surface Integrals скачать в хорошем качестве

Stokes’ Theorem Made Simple! Converting Line Integrals to Surface Integrals

Welcome back to Math and Engineering Made Easy! Today, we take an important step in vector calculus: Stokes’ Theorem. Stokes’ Theorem connects a line integral around a closed curve with a surface integral of the curl over the surface bounded by that curve: ∮ 𝐹⋅𝑑𝑟 = ∬𝑆(∇×𝐹)⋅𝑛 𝑑𝑆 ∂𝑆 In this lesson you will learn: ✔ What Stokes’ Theorem actually says ✔ Why it extends Green’s Theorem to 3D ✔ How to compute curl ✔ How to parameterize a surface ✔ How to compute 𝑟𝑢 × 𝑟𝑣​ ✔ How to convert line integrals into surface integrals ✔ Two fully solved examples, including a tilted plane and a first-octant surface This theorem simplifies many difficult line integrals, especially when the surface is easier to work with than its boundary. If you have questions, leave them in the comments — I’m happy to help! #StokesTheorem #VectorCalculus #Calculus3 #MultivariableCalculus #SurfaceIntegral #LineIntegral #Curl #GreenTheorem #EngineeringMath #MathMadeEasy