Flux Integrals, Multivariable Calculus

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We look at vector surface integrals, aka flux integrals, aka surface integrals of vector fields. We can imagine the surface S as a permeable membrane that allows a fluid or gas, represented by the vector field F, to flow through it. This analogy helps us understand the orientation and direction of fluid flow across the surface.

We start by comparing scalar surface integrals with vector surface integrals, noting the similarities in the process but also highlighting the crucial difference in how we handle the orientation of the surface. For vector surface integrals, it is essential that the surface S has a specified orientation, which is usually provided in the problem statement.

The process involves

(1) parametrizing the surface,
(2) evaluating the vector field along the parametrization,
(3) differentiating the component functions with respect to the parameters, and then
(4) computing the cross product of these derivatives.

Unlike scalar surface integrals, for vector surface integrals, the orientation of the cross product vector matters significantly as it determines the direction of the normal vector relative to the surface.

So I include a discussion on the concept of orientability, using illustrated examples to show how to analyze the orientation of a given surface. We want to only work with orientable surfaces for the computation of flux integrals, where we can make a consistent choice for the direction of the normal vector.

(Unit 6 Lecture 14)

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