Exercise 1.5.13. Suppose we want a plane curve B(s) with curvature a specified curvature function « (s). Show that such a ß may be defined by B(s) = ( | cos 0(u) du, | sin 0 (u) du

Q: By using Green's theorem evaluate £ (Ary – y") dr + (** + 3ry) dy where C is a closed curve starting…

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Q: Let r= (3v/2t, 3ť, /2t³) be a curve in 3-dimensional space having unit tangent and normal vectors u…

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Q: 04. Show that the Gauas and mean curvatures on, X-(u+v,u-v, uv) at u=1, v=1 are K=1/16 and H= JI.…

A: Given: 1) X=(u+v,u-v,uv) at u=1,v=1. 2) The paraboloid is xy=aZ.

Q: Consider points P and Q on a curve. What does it mean for the curvature at P to be less than the…

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Q: 2. Show that if a : [a, b] → R" is a regular parameterization of a curve then the curvature at α(t)…

A: Let, is a regular parameterization of a curve. To show the curvature at  is .

Q: Find the equation for the principal curvatures, the differential equation of the lines of curvature,…

A: 2∂z∂x=2xaa=xp, where p=∂z∂x2∂z∂y=2ybb=yq, where q=∂z∂y2z=x2xp+y2yq2z=px+qy

Q: a. Find a parametrization of the curve x = (y* – 3)e2y.

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Exercise (2): Calculate S. xydx + 3y*dy where C is the curve in xy plane, x = y² from (0,0). to…

A: We have to find the line integral

Q: Determine the maximum curvature for the graph of f(x) = 6 In (3x). The maximum curvature is at x =

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Q: 2. Find the coordinate of the points on the cardioid r=1- cose at which there is a horizontal…

A: We are given the polar curve: r = 1- cosθ use formula:         x=r cosθy=r sinθ then,  dxdθ=-sinθ+2…

Q: 6. Set up but do NOT evaluate Evaluate the line integral 2 curve (t) - (t", t), where 0sts0. L ds,…

A: Consider the given integral as ∫Cyxds, where C is the curve r⇀t=t4,t3 and 0≤t≤1. Note that, the line…

Q: Find the curvature of the curve r(t). r(t) = (3 + 6 cos 9t) i - (8 + 6 sin 9t)j + 6k Group of…

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Q: Compute the work W against the earth's gravitational field required to move a satellite of mass m =…

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Q: Let P be a point on a regular surface S and let f and g be two curves on S passing through P and…

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Q: Is the below statement True or False? The parametrization for a given curve is unique.

A: No. Parametrization of a curve is not unique. For example y = x2 can be written as (t,t2) or…

Q: Find the center of curvature of the ff. curves at the pt. indicated: 1.) y=x+-x² at (0,0); 2.)…

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Q: Find the derivative dydx of the hyperbolic spiral r=aθ.

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Q: Determine the curvature of the graph of y = In x at the point where x = 1.

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Q: 2-Show that the regular curve a = a(s) is a straight line if and only if its curvature k is…

A: Here we have to show that a regular curve is a straight line if and only if, its curvature is…

Q: Compute the mean curvature of the surface z = xy at the origin

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Q: 1) T(t) - (cost + 1 sint)i +(int-t cost) J +r'k %3D + rk

A: The arc rate at which the tangent changes direction as the point P moves along the curve is called…

Q: Use Theorem 11.24 to prove that the curvature of a linearfunction y = mx + b is zero for every value…

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Q: 1) Calculate the curvature of the position vector r(t) = sintāx + 2costāy + V3 sin t āz is a circle,…

A: Given position vector is  r→t=sintax→+2costay→+3sintaz→ Then unit tangent vector is…

Q: 4) Find the families of lines of curvature of the surface z = r + y? .

A: Given: z=x2+y2

Q: Exercise 1.3.2 Give a paramatrization for the curve curve y = x³ + ()x-¹, then deduce the length of…

A: Given curve is  y=16x3+12x-1

Q: use Eq. (3) to calculate the curvature function κ(t)  r(t) = (t−1, 1, t)

A: Solution:Given     r(t)=(t-1, 1,t)To find κ(t):κ(t)=r'(t)×r''(t)r'(t)3

Q: True or false: the curvature of,k, for x^2+y^2=100 is larger than for x^2+y^2=1/4

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Q: Find the center of curvature of the curve y-Inx at (1, 0).

A: We Know that Centre of curvature for y=f(x) is κ=|f′′(x)|/(1+f′(x)3/2  -----(1)

Q: Distinguish briefly the Skewness and kurtosis

A: Skewness is a measure of symmetry where symmetry means equal proportion to the left and right to the…

Q: Compute the mean curvature for u = 1 and v= 0.

A: Given,           X(u, v)=u2-cosv, u+sin t, cosv To find the mean curvature for u=1, and v=0.

Q: At what point does the curve y = e have maximum curvature? What happens to the curvature as x → ∞?

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Q: Obtain the gradient to a curve with the parametric equations given by I = In - and y = sinh 2t whent…

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Q: I) Find the extreme values of the function : &r+ 2ry-3x+y +1 and show that only one of the two…

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Q: Detemine the lineality over a full langth in the a-direston of he wave Sine Curve y- Asina

A: We know that the wave length of y=A sin x is 2π.  Note: The linearity over a full wave length of the…

Q: 2. Show that if a : [a, b] → R" is a regular parameterization of a curve then the curvature at α(t)…

A: Let,  is a regular parameterization of a curve To show the curvature at  is

Q: Explain why a cubic Bézier curve is completely determined by x(0), x (0), х(1), and x'(1).

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Q: 4. Find the equation of the tangent plane to the surface ey + In(xz) – evz = 0 at the point (1,0,…

A: Given:

Q: 4. Consider the closed curve C, which is the image of the path 7(t) = (a cos t, b sin t), t E [0,…

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Q: Exercise 3. Show that the Gaussian curvature of a surface which is home- omorphic to the torus must…

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Q: Does the the paraboloid z = 1 – 2x² - - y² have a tangent plane at some – x + y + z = 1?

A: Introduction: Let P0=(x0,y0) is a point in the domain of f, and S is a surface defined by a…

Q: Find the equation for the principal curvatures, the differential equation of the lines of curvature,…

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Q: Find a plane curve a(s) endowed with a unit-speed prametrization by SE (-0, 0) such that its signed…

A: The curvature of the curve αs is given by: αs=∫s0scos(θ(u))du, ∫s0ssin(θ(u))du-----(1) in which…

Q: Show that the Gauas and mean curvatures on , X = (u + v,u –- v,uv) at u = 1, v = 1 are K = 116 and H…

A: Given, X=u+v,u-v,uv u=1,v=1 , k=116 and H=182 We know that, E=xu,xuF=xu,xvG=xv,xv So, we need to…

Q: Show that for a plane curve the torsion T = 0.

A: We have to shown that, For a plane curve the torsion T=0

Q: .3 Find the radius of curvature of y=x' at x=1,

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Q: What is the radius of curvature at point (1, 2) of the curve 4x - y2 = 0.

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Q: Determine the maximum curvature for the graph of f(x) = 7 In (2x). The maximum curvature is at x=

A: We are given function f(x)=7ln(2x) of the graph Formula for curvature is κ(x)=|f′′(x)(1+(f′(x))2)32|…

Q: 10. Find a tangent plane to z² – x² +y² = 1 at (7, –5, 5).

A: To find a tangent plane to the given curve at the point (7,-5,5)

Q: Suppose you are told that there does not exist a smooth space curve K de- scribed by a vector…

A: In this question, we need to check the validity of the arc length function. The given arc-length…

Q: The curvature of y = x2

A: so we have to find curvature of y =  x2. we use the formulae for curvature  k(x) =…