Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,
(1)
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The first fundamental form satisfies
(2)
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The first fundamental form (or line element) is given explicitly by the Riemannian metric
(3)
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It determines the arc length of a curve on a surface. The coefficients are given by
(4)
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(5)
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(6)
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The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities
(7)
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(8)
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are called scale factors.