MHB What is the curvature of a graph at a point?

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The curvature of a graph at a point is defined mathematically for a smooth function f: (a,b) → R. At any point x₀ within the interval (a,b), the curvature is given by the formula κ = f''(x₀) / (1 + (f'(x₀))²)^(3/2). This formula highlights the relationship between the second derivative of the function and the first derivative at that point. Participants are encouraged to share their attempts to solve related problems for more targeted assistance. Understanding this concept is crucial for analyzing the geometric properties of curves.
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Consider the curve which is graph of a smooth function $$ f : (a,b) → R$$. Show that at any $$ {x}_{0}\:s.t\:{x}_{0} ∈ (a,b)$$ the curvature is $$\frac{{f}^{''}({x}_{0})}{{(1+{{f}^{'}({x}_{0})}^{2})}^{3/2}}$$.
 
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Can you show us what you've tried so far so we know where you are stuck and can offer better help?
 
Thread 'Problem with calculating projections of curl using rotation of contour'

Thread 'Problem with calculating projections of curl using rotation of contour'

Hello! I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem. Given: ##rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0## ##rot_yA=\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x}=1## ##rot_zA=\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y}=0## I rotated ##yz##-plane of this coordinate system by an angle ##45## degrees about ##x##-axis and used rotation matrix to...
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