How is the flux through a curved surface approximated using local planar segments?

Conditions

• The surface is curved.
• The surface is divided into small rectangular patches.
• The patches are sufficiently small to approximate flat planes.

Reasoning, step by step

1. Divide the curved surface into small rectangular patches.
2. Zoom in on a single small patch.
3. Observe that the patch approximates a flat plane segment.
4. Assume the normal vector $\vec{n}$ and field $\vec{F}$ are constant over this small patch.
5. Calculate the differential flux for this patch using the planar formula.

Example

A small rectangular patch on a blue mesh bowl shape is highlighted and zoomed in to show it approximates a flat plane segment.

Common misconceptions

• Assuming the normal vector is constant over the entire curved surface.
• Believing that curved surfaces cannot be approximated by flat planes.
• Ignoring the requirement that patches must be sufficiently small.

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Connected concepts

Explore next

• How is the total flux over the entire curved surface calculated?
• What is the mathematical expression for the differential flux?
• How does this approximation relate to the concept of a surface integral?