25.* Let 7(s) be a unit speed curve in R³ with nowhere vanishing curvature k. Let a > 0 and consider the tubular surface o(s, 0) = y(s) + a(cos(0)n(s) + sin(0)b(s)). Prove that if aK < 1 everywhere, then o is a regular surface.

Q: (ii)If a = a(s)is a regular differentiable curve , then show that :- (a) kt = -T · B', (b)t = B · N…

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Q: Consider a plane curve with a regular twice continuously differentiable parametrization r(t) =…

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Q: Find T, N, and x for the space curve r(t) = (6e'cos t) i + (6e'sin t)j+ 4k. T(1) = Di+Oi+k (Type…

A: Given r→t=6etcosti^+6etsintj^+4k^ We have to find the T, N and κ for the given curve. Differentiate…

Q: Use the formula x(x) = = K(x) = y = 3x4 |f"(x)| [1 + (²²(x))²] ¹ to find the curvature.

A: Given information:Function: To find:The curvature of the function .Concept used:The curvature of a…

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Q: F(x, y) = x² i + y² j C is the arc of the parabola y = 4x² from (-1, 4) to (0, 0) %3D (a) Find a…

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Q: Let C be the simple closed curve which consists of the line segment from (1,1) to (-2,4) followed by…

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Q: 1) Let S be the surface of the cylinder r+y = 4 with y >0 and -15z<1. Find a parametric equation…

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Q: 1. Find the Curvature at the Point (0,3) above the ellipse x = 4(ost, y = 3 sint

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Q: #10 For the surface x³-2xz+y4+1=xy do: (a) Find the tangent plane at point (1,1,1),, Find…

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A: Given C be the curve of intersection of the surfaces y=2x2 and z=223xy

Q: Determine ar, aN and k for the space curve r = (e' cost – e' sin t)i + ((V5)e')j + (e' cos t + e'…

A: Explanation  Given space curve = r =etcost-etsinti +5etj+etcost+etsintk we have to find - aT , aN, κ…