How is the total flux through a curved surface derived as a double integral?

Conditions

• The surface is curved.
• The surface is divided into infinitesimal elements.
• The differential flux for each element is defined.

Reasoning, step by step

1. Define the differential flux for a tiny element as $d\Phi = \vec{F} \cdot \vec{n} dS$.
2. Sum up all these infinitesimal contributions over the entire surface.
3. Recognize that this summation becomes a double integral (surface integral).
4. Express the total flux as $\Phi = \iint_S \vec{F} \cdot \vec{n} dS$.

Example

Based on the local planar assumption, the differential flux is $d\Phi = \vec{F} \cdot \vec{n} dS$, and the total flux is the integral of this expression across the surface.

Common misconceptions

• Confusing a double integral with a single integral.
• Believing that the total flux is simply the sum of a finite number of patches.
• Ignoring the vector nature of the field and normal in the integrand.

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Connected concepts

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• What is the geometric meaning of the surface integral?
• How does the orientation of the surface affect the total flux?
• Can this method be applied to closed surfaces?