A way to define curvature then would be to find the "tangent circle" (if it exists) at each point, then the curvature would be the reciprocal of the radius of this "tangent circle".
The radius of curvature is the radius of the osculating circle. Curvature is the reciprocal of the radius of curvature. Once you have a formula that describes curvature, you find the maximum curvature (or minimum radius) the same way you find the extrema of any smooth function.
In a problem to calculate radius of curvature at a point $dy/dx$ becomes undefined and hence the book considers $dx/dy$ instead and rewrites the formula for the ...
The curvature, on the other hand, is the inverse of the radius of the circle that best approximates the curve at that point, a.k.a. the osculating circle. What makes for the “best” approximation is given a precise mathematical definition in calculus.
I want to understand the basic conceptual idea about intrinsic and extrinsic curvature. If we consider a plane sheet of paper (whose intrinsic curvature is zero) rolled into a cylindrical shape, th...
The Gaussian curvature is the ratio of the solid angle subtended by the normal projection of a small patch divided by the area of that patch. The fact that this ratio is based totally on the definition of distance within the surface (independent of the embedding of the surface; that is, bending and twisting, etc.) is Gauss' Theorema Egregium.
Is there any easy way to understand the definition of Gaussian Curvature?
Yes, you are right- curvature is meaningless without Riemannian metrics or adjacent structures, which exactly introduce the idea of "curvature" 3. You are wrong in the wrong direction- not only are tangent spaces and the definition of smooth manifolds insufficient to define curvature, Riemannian metrics are too, and curvature is an even ...