✳flâneur — a map of the web's best reading

First fundamental form - Wikipedia

en.wikipedia.org · 1,317 words · saved by 1 readers

In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space. The first fundamental form is denoted by the Roman numeral I, I ( 𝑥 , 𝑦 ) = ⟨ 𝑥 , 𝑦 ⟩ . Let X(u, v) be a parametric surface. Then the inner product of two tangent vectors is I ( 𝑎 𝑋 𝑢 + 𝑏 𝑋 𝑣 , 𝑐 𝑋 𝑢 + 𝑑 𝑋 𝑣 ) = 𝑎 𝑐 ⟨ 𝑋 𝑢 , 𝑋 𝑢 ⟩ + ( 𝑎 𝑑 + 𝑏 𝑐 ) ⟨ 𝑋 𝑢 , 𝑋 𝑣 ⟩ + 𝑏 𝑑 ⟨ 𝑋 𝑣 , 𝑋 𝑣 ⟩ = 𝐸 𝑎 𝑐 + 𝐹 ( 𝑎 𝑑 + 𝑏 𝑐 ) + 𝐺 𝑏 𝑑 , where E, F, and G are the coefficients of the first fundamental form. The first fundamental form may be represented as a symmetric matrix. I ( 𝑥 , 𝑦 ) = 𝑥 𝑇 [ 𝐸 𝐹 𝐹 𝐺 ] 𝑦 When the first fundamental form is written with only one argument, it denotes the inner produ

First fundamental form - Wikipedia Jump to content From Wikipedia, the free encyclopedia Inner product of a surface in 3D, induced by the dot product In differential geometry , the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of 3</sup>"}},"i":0}}]}'> R 3 . It permits the calculation of curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space . The first fundamental form is denoted by the Roman numeral I , I ( x , y )

Explore this link on the map →

related reading