How to Compute Surface Integrals in Vector Fields | Flux Integrals, Parametric Surfaces & Examples
Автор: Math and Engineering Made Easy
Загружено: 2025-12-05
Просмотров: 4
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!
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