How to Compute Surface Integrals in Vector Fields | Flux Integrals, Parametric Surfaces & Examples скачать в хорошем качестве

How to Compute Surface Integrals in Vector Fields | Flux Integrals, Parametric Surfaces & Examples

Hello everyone, and welcome back to Math and Engineering Made Easy! In today’s lesson, we complete our discussion of surface integrals — this time working with vector fields. Last time, we evaluated surface integrals over scalar fields. Today, we extend that idea to compute flux through a surface using vector calculus. 🌟 What You Will Learn Today ✔ How to project a vector field onto a surface’s normal direction ✔ Using ru × rv to get the normal vector to a parametrized surface ✔ Why |ru × rv| represents the area scaling factor ✔ How to compute the surface integral ∬ 𝐹⋅𝑛 𝑑𝑆 𝑆 ✔ The meaning of flux in vector fields ✔ Step-by-step examples including: A surface defined implicitly (a downward-opening paraboloid) A surface already given in parametric form ✔ Parameterization, partial derivatives, cross products, and dot products all tied together intuitively This lesson contains two full worked examples, including the full evaluation of 𝐹 = 𝑦𝑖 + 𝑥𝑗 + 𝑧𝑘 over a bounded paraboloid, and a more advanced example with a provided parameterization. If you have any questions, feel free to leave them in the comments — I’m always happy to help. Thank you for watching, and see you in the next video!