To illustrate that the tength of a smooth space curve does not depend on the parameterization used to compute it, calculate the length of one tum of the helix with the following parameterizations. a. r(t) = (cos 41)i + (sin 41)j +4lk, Osts b. r(t) =cos i+ sin k Osts4x C. r(t) = (cos t)i - (sin t) - tk 2xsts0 Note that the helix shown to the right is just one example of such a helix, and does not exactily correspond to the parametrizations in parts a, b, or c

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: • Suppose that the mass in a mass-spring-dashpot system with m = 25, c = 10, and k = 226 is set in…

A: We'll answer the first question since the exact one wasn't specified. Please submit a new question…

Q: с 2 0 Use Green's theorem to evaluate the line integral _cos fc (2tan x − 2y³) dx + (ä⁰⁹ y + 2x…

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Q: Evaluate : //| cos(x² +y³) dV, where E is the region bounded by the disk x² + y? < 9 E and the plane…

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Q: Find the arclength of the graph of the fun ction over the indicated inter val. (a) y = 2x3/2 + 3, 0…

A: Given : Functions and their interval (a) y = 2x3/2 + 3 , 0≤x≤9 (b) y = ln(cosx), [0, π3] (c) x =…

Q: the surface z = &x + 4y th=

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Q: Let C be the curve in R³ parameterized by r(t) = (sin t, 2t, cos t) for 0< t< /2. (a) Compute the…

A: See the attachment

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: Fi for oec ind volume of the solid of revolution by rolahng about the -ais each regton boundad b the…

A: To calculate the volume of the given solid of revolution

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A: Length of the curve, rt ,  L=∫abrt˙dt ,    a≤t≤b (a)  Consider the parametrisation of the helix,…

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: 19. If R is the region enclosed by a simple closed positively-oriented curve C, then the area of R…

A: Given that if R is the region enclosed by a simple closed positively oriented curve C . We need to…

Q: 5. Find the volume of liquid needed to fill a sphere of radius R to height h (Figure 18). R h…

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Q: Sketch the space curve r(t) = ⟨2 sin t, 5t, 2 cos t⟩ and find its length over the given interval [0,…

A: The given equation of space curve is  r(t) = ⟨2 sin t, 5t, 2 cos t⟩.

Q: |Use Green's Theorem to evaluate | -x'ydx+xy°dy where C is the boundary of the C circle x + y =16…

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Q: 3. If C is the unit circle in the complex plane C, and f(2)= =², show that f(z) dz = 0 using two…

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Q: The region between the curve y = Vxe- and the x-axis for0 <r<∞ is rotated about the r-axis to create…

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Q: Let L be the tangent line to the hyperbola xy = 1 at x = a, where a &gt; 0. Show that the area of…

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Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: Consider the cune C: (x-8)+ 4-7) When parametneing C, A tungent veotoar is i O a) r (0) = (8 +3…

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Q: Use the theorem on saddle-node bifurcations from the lecture' to show that i = sin(r) sin(r) +…

A: Given dxdt=sinrsinx+cosx-er. Let us prove that 0,0 is the saddle-node bifurcation.   Theorem:…

Q: Sketch the space curve r(t) = a cos ti + a sin tj + btk and find its length over the given interval…

A: Given : Space curve equation rt = acosti^ + asintj^ + btk^  ,   and  t ∈ 0,2π  ,a and b are contants…

Q: Consider the following. C: line segment from (0, 0) to (5, 10) (a) Find a parametrization of the…

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Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

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Q: Verify the Green's theorem for the following integral $. sinydx + (x – cosy) dy where curve C is the…

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Q: xy = 4

A: Solution

Q: 3. By using Green's Theorem in the plane, evaluate | (y - sinx) dx + cos x dy where C is the…

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Q: 7. Given the space curve r(t) = (4 sin t,4 cos t, 10t), Find T, N, B, and T(torsion)

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Q: The volume of the solid generated by the curve y = e 2, between z = 2 and z = 6 about the a - aris…

A: Solution-

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

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Q: Suppose the divergence of the electric field in some spherical region is found to be V E = 5kr³.…

A: as given,   the divergence of electric field in some spherical region is ∇·E=5kr3 claim- to find the…

Q: For each of the following, is this a valid parametrization of the surface 2x 3y
z  4? Circle any…

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Q: (b) Consider F(x, y) = -i+ j, where (x, y) # (0, 0). x²+y2 x2+y2- Without using Green's Theorem,…

A: Consider the provided function,

Q: 2. Let R(t) = }sin 4t î – bcos“(21) j + 2t Ř. Find the moving %3D trihedral of the curve of R(t) at…

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Q: Verify the Green's theorem for the following integral O sinydx + (x – cosy)dy where curve C is the…

A: Answer

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: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A: Since you have posted multiple questions but according to guidelines we will solve the first…

Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: For point p in a regular surface S in R³, the exponential map exp p induces the normal coordinates…

A: Let M,g be a Riemann manifold and let v,u be the ormla coordinates in p∈M. For every v∈TpM denote…

Q: 29. Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given:

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

A:

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

A:

Q: . Find the solution of the following Cauchy problem ", = F(x, y) given u = f(x). du - = g(x) on the…

A: It is given that…

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

A: First we have to find partial differentiation with respect to x , y and z of function f such that-

Q: To illustrate that the length of a smooth space curve does not depend on the parameterization used…

A: To find - To illustrate that the length of a smooth space curve does not depend on the…

Q: 1. Find parametrizations of the curve of intersection of x² + 4x + z² = 0 and x- – 2y + z = 3.

A: Given curve is x2 + 4x + z2 = 0 .  This can be written as (x+2)2 + z2  = 4  So the curve is a circle…

Q: Let C be the curve in the plane with parametrization r(t) = cos ti+ sin tj for t e [0, 4T]. Compute…

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Q: Covider that te curve c is an inters ecdios of the surfaces SMown below If Cirlt)= (x(t), (t), Zlt)…

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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.