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Appologies for formatting issues this is the first time I have submitted something to the forum.

I have a pretty simple problem, I am just going through the derivation of the First Fundamental Form and I think I am missing someting in the derivation.

If we have a point x = (x1,x2) on the 2D surface ω(x) then the derivative matrix of the surface can be defined as Dω(x) = (∂ω(x)/∂x1,∂ω(x)/∂x2) which is 3x2 matrix.

| 1 0 |

| 0 1 |

| ∂ω/∂x1 ∂ω/∂x2 |

thus the first column is the partial derivative of the surface with respect to x1 and the second column is the partial derivative with respect to x2 and are thus the non unit tangent vectors.

The first fundamental form can be defined as I by taking the inner product of the two tangent vectors :

I = (Dω(x))' Dω(x).

which can be written as:

|1+(∂ω/∂x1)^2 (∂ω/∂x1)*(∂ω/∂x2) |

|(∂ω/∂x1)*(∂ω/∂x2) 1+(∂ω/∂x1)^2 |

which can be re-written as the metric tensor:

|E F|

|F G|

However looking here:

http://mathworld.wolfram.com/FirstFundamentalForm.html

they appear to have dropped the 1 in E and G. I was just wondering where I might have gone wrong?

Many thanks in advance

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# First Fundamental Form from tangent vectors

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