Chapter 2 - Exercises 1) Calculate the curvature of the position vector f(t) = sin tāx + %3D 2cos tāy + v3sin tāz is a circle, also find its radius.

Q: Find the curvature of r(t) = at t = ,.

A: Given: r→t=3cost, 3sint, t Calculation: Compute the Curvature at t=π2 as follows. It is known that…

Q: Find the point where the helix r(t) = (cos(t), sin(t), t) intersects the plane z = (х, у, 2) %3D

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Q: Describe how the surface with spherical equation p²(1 + A cos² ¢) = 1 depends on the constant A.

A: Given : ρ2(1+Acos2ϕ)=1To describe: How the surface with spherical equation  ρ2(1+Acos2ϕ)=1depends on…

Q: Find the curvature and radius of curvature of the plane curve at the given value of x . y = x −4/ x,…

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Q: Show that the ellipse x = a cos t, y = b sin t, a 7 b 7 0, has its largest curvature on its major…

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Q: Prove that the parabola y = x^2 at the points where the line has an angle of inclination "o" the…

A: Curvature of  a graph y= f(x)  k=d2ydx21+dydx232

Q: At how many point(s) does the helix with vector equation r(t) : (cos t, sin t, t) intersect the…

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Q: 2.Let a be a unit speed curve lying on a sphere center c and radius a > 0. Find the minimum value of…

A: Given That : Let α be the unit speed  curve lying on a sphere center c and radius a>0. To Find :…

Q: 8) If curvature of a space curve at (2,3) is 8 and curvature at (7,1) t=7 is 1, what can be said…

A: ANSWER:

Q: Chapter 13, Section 13.7, Question 017 (a) Find all points of intersection of the line x = -1+t, y…

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Q: A curve is given in polar coordinate as r=R(f). What is its curvature? Find it in terms of R,R' and…

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Q: The vector equation r(u,v)=ucosvi+usinvj+vk, 0≤v≤4π, 0≤u≤1, describes a helicoid (spiral ramp). What…

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Q: Sketch the set (described in cylindrical coordinates). r = 4

A: GIVEN r=4 we need to sketch the cylindrical coordinates

Q: 2) The (Beltrami) pseudo sphere is given by the chart xlu,v) = (cosculsinlv), sincu) sincv) , cos(vl…

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Q: Exercise 9.6. Find the average distance of the space between a sphere of radius 2 and a sphere of…

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Q: In 1932, Giuseppe Momo was commissioned to build the famous Vatican Museum double spiral staircase.…

A: Given information: The radius of the spiral is 29 feet and height is 60 feet that makes 6…

Q: The curvature of r = 1+ cos 0, 0: 2 2/2 5 2/2 3/3 2 3/3 3.

A: Given: r=1+cosθ,θ=π2 κθ=2f'θ2+fθ2-fθf"θf'θ2+fθ232   r=f(θ)r=1+cosθ f'θ=-sinθf"θ=-cosθ

Q: Exercise 9. Show that on a simply connected surface of negative curvature two geodesics emanating…

A: We will suppose that further the Gaussian curvature of the surface is negative except for at the…

Q: 1. (a) Parametrize the curve of intersection of the sphere a? + y? + 22 22 + 222 = 1, Try your best…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Write a formula for the curvature of a plane curve with equation y = f(x). %3D

A: If the equation y=fx. Then the formula of radius curvature at the point is:   Rx=1+dydx232d2ydx2…

Q: What are the rectangular coordinates of the point whose cylindrical coordinates are (r = 7,0 = 1.2,…

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Q: Exercise 7. Show that the curvature of a circle of radius r is , and the curvature of the line is…

A: Show that the curvature of a circle of radius r is 1r and the curvature of the line is zero.

Q: Exercise 5. Show that there exists at most one closed geodesic on a cylinder with negative…

A: We will assume S ⊂ R be a surface homeomorphic to a cylinder and with Gaussian curvature K < 0.…

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: A curve has a vector equation r = ti+ 4 sin(t)j + 4 cos(t)k. Find the curvature, K.

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Q: Find the center of curvature of y = x3 at the point (1, 1) A. (-4, 8/3) C. (6, -2/3) B. (4, -8/3)…

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Q: the Cartesian coordinates of the point with cylindrical coordinates (r, 0, 2) = (4, т, 6) (x, y, 2)…

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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: are the rectangular coordinates of the point whose cylindrical coordinates are (r = 6, 0 = 2, z = 6)…

A: To convert cylindrical coordinates to rectangular coordinates.

Q: Just like in a projective plane, two distinct lines always intersect on the spherical plane.

A: Straight lines in spherical plane are the great circles. So, any two different lines meets at two…

Q: Write a formula for the curvature of a plane curve with equation y= f(x)

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Q: Find the curvature of r(t) = at the point (1,0,0)

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Q: Find the polar coovdinates rectanguar coordinak es the point whose (x,y)= LI,}) are

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Q: 2. Convert the point (V6, , v2) from cylindrical to spherical coordinates. (e) (2VZ,품, 끌) (d) (vZ )…

A: The solution is given as

Q: The curvature of y=x? is:

A: The curvature of a curve is defined as the amount by which a curve deviates from being a straight…

Q: Show that the parametric equations x a cosh u cos v, y b cosh u sin v c sinh u represent a…

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Q: Find the Cartesian coordinates of the point (6, 4. VIT) given in cylindrical coordinates

A: First we have to determine (r,∅,z) then calculate respective Cartesian co-ordinates

Q: Convert the point (1, , V3) from cylindrical to spherical coordinates. A. (2. 5. 5) 4' 6 В. 2,

A: To convert cylindrical coordinates (r, θ, z) to spherical coordinates ( ρ, θ, φ) we use following…

Q: 63. х 3D 2t, у %3D 31, 0 <t < 3 (а) х-ахis (b) у-ахis

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Q: Exercise 2. Show that the total geodesic curvature of a simple closed planar curve is 27.

A: We will use the Tangent-Normal frame to prove the given result.

Q: Show that the ellipse x = a cos t, y = b sin t, a &gt; b &gt; 0, has its largest curvature on its…

A: Consider the given equation as x=acost, y=bsint and a&gt;b&gt;0

Q: 8) If curvature of a space curve at (2,3) is 8 and curvature at (7,1) t=7 is 1, what can be said…

A: Given that curvature of a space curve at 2,3 is 8 and curvature at 7,1 is 1.

Q: 4. Find the curvature of r(t) = (tv2, e', e-t) at the point (0,1,1).

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Q: If 7(t) = then the curvature at the point (2,0,0) is

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Q: Find the curvature of rt) = Point (2,0, "2) P"(t)=(2Binf-9 Coo(3t), 0> k(4.0%)

A: Given rt=2sint, cos3t,t at point 2,0,π2r't=2cost, -3sin3t,1r''t=-2sint, -9sin3t,0

Q: Show that the ellipse x = a cos t, y = b sin t, a&gt; b&gt;0, has its largest curvature on its major…

A: The curvature for a parametrized curve c(t) = x(t)i + y(t)j is :

Q: What is the radius of curvature of the polar curve r = 02 at 0=3/2?

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Q: 1) Calculate the curvature of the position vector 7(t) = sin tax + %3D 2cos tay + V3 sin tāz is a…

A: In this question we have to find curvature and radius of curvature.

Q: Theorem: Two geodesics at right angles have their torsion equal in magnitude but opposite in sign.

A: To prove: Two geodesics at right angles have their torsion equal in magnitude but opposite in sign.…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.