Exercise 3. Show that the Gaussian curvature of a surface which is home- omorphic to the torus must alwasy be equal to zero at some point. Exercise 4 Show that simple closed curve with total geodesic curature

Q: For any curve in the plane, the curvature is the absolute value of the geodesic curvature. * True…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: Find the curvature of y = sin(3T) at a = =.

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

Q: Does the Divergence Theorem apply to surfaces that are not closed?

A: By Stokes' theorem:

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

A:

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

A:

Q: Find the curvature and radius of curvature of the plane curve at the given value of x.

A: We have,   It is known that radius of curvature is given as, Here K is curvature of the curve.…

Q: Show that the parabola y = ax2, a ≠ 0, has its largest curvature at its vertex and has no minimum…

A: Using the relation,

Q: (b) Find the circle of curvature at (0, 1)of the curve y = x3 + 2x² + x +1

A:

Q: Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive…

A: We will use the Gauss–Bonnet theorem to prove the given statement.

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: Find the curvature of y = sin(3x) at a = 프 4

A: We Know that The curvature measures how fast a curve is changing direction at a given point.

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: a) Find a plane curve whose signed curvature is given by k,(s) where a is a positive constant and s>…

A: The given signed curvature is ks(s)=12as

Q: For a point moving with uniform circular motion, the distance traveled per unit timeby the point is…

A: "For a point moving with uniform circular motion, the distance traveled per unit timeby the point is…

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

A: WE HAVEDB/DS = τNWHERE AT ANY POINT P[X,Y,Z] ON THE CURVEB IS UNIT VECTOR CALLED BINORMAL TO THE…

Q: . Sketch the curve parametrized by r(t) = (ltl + t, ltl - t)

A: The given curve is rt=t+t, t-t. Sketch the graph of the above curve as shown below.

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 curvature of the plane curve with equation y (x, f (x)) on the curve, by using the…

A: We will find out the required value of curvature .

Q: Find the radii of curvature and torsion at a point of the curve    x^2+y^2=a^2 , x^2-y^2=az

A:

Q: The radius of curvature of the curve r = 4cos 20 at 0 = is %3D 4

A:

Q: Let V be a connected neighborhood of a point p of a surface S, and assume that the parallel…

A: Let the two points are p and q p, q∈S And the two curves are: α:I→S β:I→Sα0=p=β0αt1=q=βt2 And the…

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: 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: |x'y" -y': y'a - y'= es K(t) = (x2 + у'?

A:

Q: Exercise 10. Show that the curvature of a planar curve which satisfies the equation y = f(r) is…

A: To Show- The curvature of a planar curve, which satisfies the equation y=fx is given by…

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

A:

Q: Use the curvature formula to find the following parameterized curve

A: Solution: The objective is to find the curvature

Q: 8. Find a plane curve a(s) endowed with a unit-speed prametrization by S E (-∞, 0) such that its…

A: Question: Find a plane curve αs endowed with a unit-speed parametrization by s∈-∞,∞ such that its…

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: 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: 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…

A:

Q: Let S be the surface obtained by revolving the plane curve 2x = /9 – y² about the y-axis.

A:

Q: Which type of surface does the curvex= cos(3t),y= sin(3t),z=t, lie on?

A:

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

A:

Q: Find the curvature and the radius of curvature at any point on the curve y = In sec x.

A: A curve given y = ln (secx)  Find out the curvature and the radius of the curvature of given curve.

Q: Determine the projection of the following surface on the yz plane. 3x + = 1 O Parallel Lines O…

A: Topic:- projection of a surface

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

A:

Q: Find the curvature of a superellipse defined by the parametric equations…

A:

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 or rthoganal tajeclolies. ! d faly Curver r= 4C cose, where C be being parameer

A:

Q: Find the curvature of the plane curve y = -t at the point t = 2. к(2) -

A: Given:y = -t4differentiating with respect to t, we get:            y' =     d(-t4)dt…

Q: In each of the following cases, define a parametrised curve y : [a, b] → C whose image * is: (a) The…

A: When we deal with contours ,we think about function of two and three variables. Contours in two…

Q: Find the length of the circle of radius r defined parametrically by x=rcost, y=rsint,0≤t≤2π.

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

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

A: The answer is given in handwritten format --

Q: If the infinite curve y = eSX, x > 0, is rotated about the x-axis, find the area of the resulting…

A:

Q: 10.) At what point does the curve have maximum Curvature? y=In(x). K(x)= f"|

A: firstly find f, f' and f'' . And substitute all these values in given curvature formula.  After that…

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

A:

Q: Let r (t,t,t) be a curve in 3-dimensional space having unit tangent and norma V2 vectors u and n.…

A: We will use formula of curvature of a curve to solve it.

Q: Let M be the surface given by z = y^2 − x^2. Compute its Gaussian and mean curvatures at all points

A:

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: 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: Find the following for the hyperboloid of one sheet (a) First and second fundamental forms (b)…

A:

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: 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: Find the curvature K of the plane curve y 3 at x = 3. 2х + K =

A:

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: Determine the radius of the curvature of the plane curve given by y In sec x at the || point (0,0).

A:

Q: What is a flat surface that extends infinitely in all directions?

A: Plane.

Q: Find the gaussian and mean curvature functions of the surface m given by the equation y3=y1y2 in R³…

A: First parameterized the equation.

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: At what point does the curve have maximum curvature? y = 7 In(x) (x, y) = What happens to the…

A: Given,              y = 7 ln(x)

Q: Consider the plane curve parametrized by F(1) -i+ (In(com(1)J, Find curvature s(4).

A:

Q: The curvature of y = x2

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