Physics I / Surface Integrals
Practice question · Fill in the blanks

Complete the following passage by choosing the correct terms from the word bank.

The magnitude |rᵤ ×r\times rᵥ| gives the local ______ of the surface element, and its direction gives the surface ______.

Word bank: area · normal · tangent · curvature

Hints
  1. The two tangent vectors come from differentiating the parametrisation.
  2. Their cross product supplies both direction and area scaling at once.
Show the answer

The magnitude |rᵤ ×r\times rᵥ| gives the local area of the surface element, and its direction gives the surface normal.

Why

The cross product of tangent vectors gives the normal with magnitude = area element.

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