6. Find the curvature of r(t) = ti + t2j +t k. at r = 0. 7. Find the curvature of the plane y = -cos() + e 8. Find the maximum curvature of y In . nack cuwerune, 9. Find the tangential component aT and normal component aN for 3ti tj + tk. r(t) 10. Let a(t) = 2t i + e j + cos (t) k denote the acceleration of a movir v(0) i+2j- k, find the particle's velocity v(t) at any time t. . (a) Find the domain of f(x, y) =n V2 x (b) Sketch the graph of f(r, y) = 6 2y. 12. Find the limit of show it does not exists. (a) lim (xy)+(0,0) +y8 (b) ry y lim (a)(1,0) (1)2+y2

Q: Find the curvature of r(t) = (In t, 2t, t²) at (0, 2, 1).

A: Given r(t) = <ln t, 2t, t2> To find the curvature (k) at (0, 2,1)

Q: 1. Use the Stoke's Theorem to evaluate y'dx + xy dy + xz dz, C where C is the bounding curve of the…

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: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

A:

Q: Find the curvature and radius of curvature of the parabola y=x² at the origin. O K= 2, p=1/2 O k=2x,…

A:

Q: Consider the superparabolas f, (x)=x", where n is a positive integer. a. Find the curvature of the…

A: given fnx=x2n, where nis positive integer. (a) find the curvature of the graph of fn , for n=1,2,3…

Q: Find the curvature K of the plane curve at the given value of the parameter. r(t) = ti + j, t = 2 K…

A:

Q: 4. Consider the curve in the plane defined by r3+y3 = 6xy. %3D (a) Find for any point (r, y) on the…

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: ANSWER:

Q: Find the curvature of the space curve. r(t) = -10 i + (t + 4) j +(In(cos t) +9) k %3D OK= sect K =…

A: Formula of radius of curvature:      k=r→'(t)×r→''(t)r→'(t)3 First we need to find r→'(t),r→''(t),…

Q: Find the curvature and the radius of curvature at the stated point for r(t) = 3cos t ax + 4sint ay…

A: Given function is rt=3cost ax+4sint ay+ t az To find curvature and radius of curvature

Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

A:

Q: Find the curvature K of the plane curve at the given value of the parameter. r(t) = 8ti – 4tj, t = 1…

A:

Q: Find the center of curvature of the curve y =-(e*+e) at (0, 1). (А) (0, 2) в) (1, 1) с) (0, 3) D (0,…

A:

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

A: Given y =e^(-x/4)  ,x=8 Now we have to find curvature and radius of curvature at x=8

Q: 4. Let r(s) be a unit speed curve with a curvature of zeroc fo. prove that the image r lies in a…

A:

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: Find the curvature and the radius of curvature at the stated point for r(t) = 3cos t ax + 4sint ay +…

A:

Q: Find the curvature and radius of curvature of the plane curve at the given value of x. y = sin 2x, x…

A: Consider the provided equation, y=sin2x,   x=π4 Find the curvature and radius of curvature of the…

Q: 4. Consider the saddle surface r y?- z², find the equation for the tangent plane at the point (0,1,…

A:

Q: b.) show that brnormal of the muolute the unit *-ズ+(c-s)で g =dG) |k-s) K| K* where K is the…

A:

Q: Find the curvature of r(t) = √√7ti + etj + e-k at point P(0, 1, 1).

A:

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

A:

Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

A:

Q: Consider the set S given by S = {(x, y, z) ER³ : 2 = 2x² - y²}, where A E R\{0} is constant.…

A: As per the question we are given a quadratic surface as : S = {(x,y,z) ∈ R3 | z =2x2 - λy2} And we…

Q: 1. Find the equation of the circle of curvature for the following curves: 1.1 x+y =x + y +x at…

A:

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

A:

Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

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

A: Green's theorem in a plane: If D is a region inclosed by a closed curve C then ∮C u dx+v dy=∫∫D…

Q: Find the curvature and the radius of curvature at the stated point for r(t) = 3cos t ax + 4sint ay…

A: Given, rt=3costax+4sintay+t·az Find the curvature and radius of curvature at the point t=π2.

Q: Find the curvature K of the plane curve at the given value of the parameter. r(t) = 6ti – 8tj, t = 1…

A: The formula for the curvature is κ(t) = r't×r''tr't3

Q: Find the curvature K of the plane curve at the given value of the parameter. r(t) = 4ti – 6tj, t = 1…

A:

Q: 2. Let a(t) = (3t-t, 3t2, 3t + t), for te R. %3D (i) Calculate the equation of the osculating plane…

A: Introduction: The formula of curvature of and torsion is given by, κ(t)=α"(t)…

Q: 3. Find the moving trihedral and the curvature at any point of R(t) = (In(sect), t/V2,t/V2). %3D

A: Introduction: In differential geometry, the Frenet–Serret formulas define the kinematic properties…

Q: Find the curvature of r(t) = (t, 5, t²): at a general point, and at (1,5,1).

A: Given r→t=t,5,t2 and point is 1,5,1. We have to find the curvature of r→t=t,5,t2 at point 1,5,1 The…

Q: Find the curvature and the radius of curvature at the stated point for r(t) = 3cos t ax + 4sint ay…

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

Q: 7. Consider the Surface S:z = 52² + 5y?. An equation of the plane tangent to Sat the point P (3, 3,…

A:

Q: Find the curvature, torsion and tangent plane for the curve C : x = t sin(2t); y = t cos(2t); z = 2t…

A: We have to find the curvature, torsion, and tangent plane for the givne curve.

Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

A:

Q: Find an equation for the tangent plane to the surface z = 3y^2 − 2x^2 + x at P(2, −1, −3). Express…

A: Let's find equation of tangent plane.

Q: 3) Consider the curve C which is the intersection of the surfaces shown in the attached figure. If…

A:

Q: Consider the paraboloid paremterized by o(u, v) = (u, v, u² + v²). 1. Find the second fundamental…

A: Consider the given equation of circle, γt=sint,cost,1 The formula for normal curvature is defined…

Q: Find the curvature of r(t) = 5(cos t) i + 12(sin t) j + tk at t = m/2. Ok = 8/13 Ok = 5/13 O k =…

A:

Q: find the curvature of the plane curve at the given value of the parameter. r(t)=4ti-2tj,    t=1

A: To find- curvature of the plane.

Q: Find the curvature of r(t) at the point (1, 1, 1). r(t) = (t, t², t³) k=

A: Given: vector valued function r(t)=<t, t2, t3>To Find: curvature at t=1

Q: 12. Let H be the hyperboloid where x² – y² + z² = 6. Find an equation for the tangent plane to H at…

A: As you asked only solution for Q12 so, I answered here only Q12

Q: Given x = t, y = 1/t, find the curvature at t = 2. 16 17√/17 O 16 17 O 16/17 17 16 √17

A:

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: For a plane curve r(t) = (x(t), y(t)), |æ'(t)y" (t) – x"(t)y' (t)| (x'(t)² + y' (t)²)³/2 - k(t) Use…

A: Here we have to use this equation to compute the curvature at the given point.

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.