Let V be a connected neighborhood of a point p of a surface S, and assume that the parallel transport between any two points of V does not depend on the curve joining these two points. Prove that the Gaussian curvature of V is zero.

Q: 5. Give an example of a curve C C xy plane so that its curvature as a space curve is different from…

A: The given: A curve  in the -plane defined by parametric equations and .The objective: To…

Q: Let L be the line segment with endpoints at (1,2,3) and (-1,1,4). A light source at is at the point…

A: We will find shadow of given points on the plane given and then find parametric equation of the line…

Q: Let C be the unit circle in the xy-plane and N a unit outward point normal vector of C. If / (F N)…

A: We have to choose correct option.

Q: Let r (t ) = ti + costj + t? k be a space curve. Which of the following vectors is parallel to the…

A: Solution:

Q: QI. A curve in space is represented as a function of a parameter, t, by the position vector r(1) =…

A:

Q: Given the vector function *(t) = (t, cos (2t)) Calculate the curvature when t=5.4

A:

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…

A:

Q: 2) Obtain the Curvature, the unit Tangent, Normal and Binormal Vectors of the following vector…

A:

Q: 3. Determine the curvature of r(t) =(√2 t, e²,e-).

A:

Q: The curve y = sin(πz), with 0 ≤ z ≤ 2, on the yz-plane is revolved once about the z-axis. Find a…

A: Given that y=sin πz and 0≤z≤2 The objective is to find the vector equation. The image formed is like…

Q: EXERCISE 2.8 A helix is defined by x = cy, y = Lsin(ky), -Lcos(ky), where c, L, and k are constants.…

A: Answer :-

Q: alculate the curvature of the vector function when t=0.1: 7(t) = (e", V2t, et)

A:

Q: R(t) is a smooth vector: Given R(0) = R'(0) = R''(0) = a. What is the curvature  R at t=0 b.…

A:

Q: Draw a picture of two spheres of different sizes being tangent to each other at exactly one point P…

A: For part 1 we are given two spheres of different sizes i.e different radius which are tangent to…

Q: Suppose C is a constant vector. Let F(x, y, z) = C find the flux of F through a surface S on plane…

A:

Q: 1 - Find a vector equation and parametric equations for the line segment connecting P(a, b, c) and…

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: 5. Sketch the vector-valued equations r(t)=2costi+3sintj. Then, find the curvature of the ellipse at…

A:

Q: Show that if u = u₁i + u2j + uzk is a unit vector, then the arc length parameter along the line r(1)…

A:

Q: A particle moves on a fixed surface in F.(0), which is defined by z = 2xy. During the motion, the x…

A:

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

A:

Q: find the principal curvatures and principal vectors of the saddle surface M:z = xy at the origin.

A: To find the principal curvatures and principal vectors of the saddle surface M: z = xy at the…

Q: Find the curvature k(t) of the vector function r(t) at the indicated time. r(t)= (acos(t), bsen(t),…

A:

Q: 1. Give the equation of any plane which is has a normal vector n = 2i- j + 3k.

A:

Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: Test the

A:

Q: 2. Consider the following vector function: r(t) = = a cos wti + a sin wtj + bwtk, where a, b and w…

A: We need to find radius of curvature.

Q: Give the length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in…

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: Q. find the curvature K of each plane curve traced out by the vector function r=r(t) = (3t – t3)i +…

A:

Q: 1. Consider the position vector r(t) = (t³) i + (3t) j + 4k and at t = 1 find the: (a) curvature,…

A: The general form of the parametric equation of the curve in three dimensional space is given by ,…

Q: Use the theorem given below to find the curvature of r(t) = √√√6t² i + 2tj + 2t³ k. Theorem: The…

A: Curvature is a measure of how much a curve deviates from being a straight line. It describes the…

Q: A body rotates with a constant angular velocity w = w,î + w2ĵ+w3k about an axis. Let ř be the…

A: Given that a body rotates with a constant angular velocity w= w1i+w2j+w3k about an axis. Let r be…

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.