Exercise 3.2.10. Using K = (In - m²)/(EG – F²), show that the Gauss curvature of the R- sphere S²(R) is 1/R².

Q: - Compute 5y dr a dy, where C is the arc of the parabola y = from (-2, 4) to (0, 0).

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Q: Please find: a. Curvature k and evaluate k(0). b. Torsion t and evaluate t(0).

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Q: Find the area between a large loop and the enclosed small loop of the curve  r = 4 + 8 cos(3θ).

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Q: ne if the curve (In t. t In t. 1) is tangent to the surface rz- uz? + cos Tu =1 at (0.0. 1).

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Q: 6. Find the curvature k at t = 1 if r (t) = 3tỉ +2t²j +k.

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Q: Find the length of each of the following curves: %3D %3D x (ey + e) from y = 0 to y = 1

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Q: 1. Given the space curve r(t) = (2e¹,e²¹.e^¹), find the curvature at t = 0

A: I am attaching image so that you understand each and every step.

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Q: Compute the arc length function L = ||r'(u)|| du for r(t) = (9t2, 3t2, t), and a = 0.

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Q: Let L be a tangent line to the hyperbola xy=4 at x=9 Find the area of the triangle bounded by L and…

A: Compute the tangent line as follows.

Q: 1. Evaluate SSSE z dV where E lies between the spheres x2 + y? + 2² = 1 and a? + y² + z² = 4 in the…

A: Given, ∫∫∫EzdV where E lies between the spheres x2+y2+z2=1 and x2+y2+z2=4 in the first octant.

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Q: Let c(1) be a path and T the unit tangent vector. What is / T- ds? c'(1) The arc length of e. The…

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Q: (2) Use the transformaiton u = xy, v = x² – y? to find JJ (æ* – y*)e*³ dA where R is the region in…

A: Given: The region bounded by the hyperbola xy=1 &3, x2-y2=3 &4 Where u=xy and v=x2-y2

Q: Evaluate ſ[ſp z dV where E lies between the spheres x² + y² + z² = 1 and + y*¨+ 2² = 4 in the first…

A: Solution:

Q: Given: vector-valued function Ř(t) = (3 cos 2t, 9t, −3 sin 2t) 1. Reparametrize R using the directed…

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Q: Evaluate the integral | (3x – 2 y + :)ds where C is the curve, C, and C, as shown. 3- y² + z² 2. 3…

A: Here, Curve C is equivalent to c1+c2. The point of intersections of c1 and c2 is at 0,1,0.

Q: 1. Find the area of the part of the planer+ 2u + 3z = 1 that lies inside the cylinder x2 + y? = 3.

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Q: For KEIRT and TEIR construct Co-Curve Y: IR→IR³ with constant Curvature K and constant torsion T

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Q: 4. Given YZ tangent to oJ at point Y and m<WYZ = 104, %3D what is the measure of WXY?

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Q: Use the Midpoint Rule with N = 10, 20, 30, and 50 to approximate the given curve's length. %3D c(t)…

A: We are given the curve :c(t)=(cos t, esint) The length of the curve: s=∫02π-sin t2+(cos t esin t)2…

Q: # 4. Evaluate the line integral | (x +2x²y) ds, where C is the upper half of the unit circle x2 + y2…

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Q: Use either form of Green's Theorem to evaluate the following line integral. $9x²1² 6 C $9x³y² dx +…

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Q: 1. Find the moving trihedral and the curvature at any point of the curve R(t) = cost î + sint j + k.…

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Q: Evaluate the line integral | H. dř; Ħ = e" î + [xe" + In(z)] ĵ + 22 k, and C is the path defined by

A: Given that  H→=eyi^+xey+lnzj^+2yzk^      (1)  and r→=t,1,1, t∈0,1                (2)      from…

Q: 4. Use the arc length formula to show that the length of a linear function f(x) = mx +c on the…

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Q: 45. If A = (y – 2x )1 + (3x + 2y)j, compute the circulation of A about a circle C in the xy plane…

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Q: Q1: Consider the curve in E' parameterized by C:(t) = 1- Find the arc-length of the curve r(t)? 2-…

A: Given equation of the curve is rt=cos5t, sin5t, 12t

Q: 3. Let a = a(s) a regular unit-speed curve and p(s, v) = a(s) + vT(s), where T(s) = a'(s) %3D Find K…

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Q: Solve the Laplace equation for a rectangular plate subject to the fol- lowing boundary conditions

A: Given: To solve the given Laplace equation for a rectangle plate subjected to the following boundary…

Q: Find the sope of the tangent ine tothe Given polarcove at thepointspecifiedby the valbe Of e 2.

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Q: What is the area of:  To the right of Z = 0.29 and to the left of Z = -1 Show the solution and…

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Q: 1 Evaluate dV, where E lies between the spheres x2 + y2 + z2 = 9 and x2 + y? + z2 x2 + y? + z2 36 in…

A: we want to find the triple integral ∫∫∫E1x2+y2+z2  dVwhere E lies between the sphere x2+y2+z2=9  and…

Q: The area enclosed between the curves p(a) =2 +*+ 3 and g(@) = -22 - * + is O 16 47 %3D 18 49 O -49…

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Q: The arc length re-parameterization of the line l: F(t) = (1,1, 1) + t(1,2, 3) with 0 <t < 1 is given…

A: This is a problem of curve parametrization.

Q: #4

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Q: let rit)= demti+ dsintf+? Find the length of this curve from t=0 to t = 4.

A: we have to find the length of the curve

Q: Compute the length of the curve r(t) = 8t cos(t), 8t sin(t), 2t over 0 st < 2n. Hint: V? + a2 dt = +…

A: Given

Q: Let YR be the arc of the circle {|z| = R} that lies between the ray Arg(2) = T/4 and Arg(z) = 37/4.…

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Q: Which one of the following is the parametric equa- tions of the line that is normal to the surface…

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Q: 10. Find the curvature K of the given curve. r(1) = ti +2sin(t)j+2cos (1)k

A: Curvature κ of the given curve r→(t) is T→'(t)r→'(t) where T→(t) = r→'(t)r→'(t).

Q: Find a parametrization of the curve x^(2/3)+y^(2/3)=8^(2/3) and use it to compute the area of the…

A: Finding the parameters:

Q: et +e- find the length of the curve h(x) = from r = 0 to z = In 2.

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Q: 3 dz where C, is the top half of circle with radius (b) Compute the upper bound of 2,

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Q: 2. Let r(t) = (t- sint,1-cost, tsint) Find the exact value of the curvature at t=0

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Q: Consider the parametric curve: X = = 7et y = 3t5 Which integral gives the arc length of the curve…

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