Hi,
I wonder why in the Gaussian curvature formula we have a double integral? The definition says we calculate "angular excess" of a triangle [imath]T[/imath] by adding up gaussian curvature over interior of [imath]T[/imath]. That Gaussian curvature is defined by [imath]k(p)dA[/imath] hence I would calculate the angular excess by: [math]\sum_{i=1}^{n} k(p)dA[/math] which in the limit gives [imath]\int_{T} k(p)dA[/imath] However for some unknow reason this is defined as [math]\int\int_{T} k(p)dA[/math] So why the double integral?
Thank you.
I wonder why in the Gaussian curvature formula we have a double integral? The definition says we calculate "angular excess" of a triangle [imath]T[/imath] by adding up gaussian curvature over interior of [imath]T[/imath]. That Gaussian curvature is defined by [imath]k(p)dA[/imath] hence I would calculate the angular excess by: [math]\sum_{i=1}^{n} k(p)dA[/math] which in the limit gives [imath]\int_{T} k(p)dA[/imath] However for some unknow reason this is defined as [math]\int\int_{T} k(p)dA[/math] So why the double integral?
Thank you.