Calculate the length of the curve č: [0; 1] → Rgiven by ( 글2 + 2t st2 + 2t + 5 (t):: HIN

Q: Verify that the following pairs of curves meet orthogonally. a. x2 + y2 = 4,   x2 = 3y2  b. x = 1 -…

A: Given:Verify that the following pairs of curves meet orthogonally.

Q: Let C be the closed curve indicated in the diagram. Use Green's Theorem (2,1) (o,i) (x² + y?, y³ +…

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Q: Find the length of the curve. r(t) = 3ti + 8t3/2j+ 12t2 k, 0 sts 1

A: The given curve is: r(t)=3t i+8t32 j+12t2 k ;                0≤t≤1

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

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Q: Consider the path r(t) = (4t,2t 2 ,2lnt) defined for t > 0. Find the length of the curve between the…

A: Given the path r(t) = (4t, 2t2, 2ln t), t>0. Find the length of the curve (4, 2, 0) and (20, 50,…

Q: Let C be the curve y = 8/a for 2.1 < x < 3.8.

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Q: Sketch the plane curve r(t) = t2i − 4tj and find its length over the given interval [0, 5] .

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Q: Let C be the curve in space given by the following parameterization. Then d + 2dy +adz is equal to…

A: Given ∫xdx+2dy+zdz = ∫(xi+2j+zk).(dx i + dy j + dz k) = ∫F.dr Here F = (xi+2j+zk)…

Q: Show that if P lies on the intersection of the two curves x2 - y2 = c and xy = d (c, d constants),…

A: Given : The two curves x2 - y2 = c and xy = d intersect at point P where c and d are constants.…

Q: Sketch the plane curve. r(t) = 2ti – tj, [0, 2] y y 2. 1 1 -1 1 4 -1 1 4 5 -1 y -1 2 3 4 1 2 3 4 5…

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Q: Sketch the plane curve r(t) = t3i + t2j and find its length over the given interval [0, 1] .

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Q: Find the are length of the polav Curve r=3° from o-0 to 0= 1

A: Solve the following

Q: 1. Show that the curve a(t) = (t,t² + 1, t – 1) is regular.

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

Q: Suppose I → R³ is a regular curve -c3 with zero curvature anywhere zero, and then the H-curve holds…

A: A given regular curve with zero curvature is -c3 We change in terms of t, then the regular curve is…

Q: Find the T(t), N(t), B(t), and K(t) of the following space curves:

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Q: Find the curvature K of r(t)= 2ti +t²j - k 3

A: We know that, the expression for the curvature; kt=r→'t×r→''tr→'t3 Given r→t=2ti→+t2j→-13t3k→

Q: Compute the length of the curve r(t)= (-3t, 3t + 7,–7t + 1) over the interval 0 <t <7

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Q: The line L(t) = (4 - 3t, 13 – 3t, 4t – 10) intersects the plane 2z - (4x + 2y) = 16 at the point |…

A: We have to given  line L(t)=&lt;4−3t,13−3t,4t−10&gt; which intersects the plane 2z−(4x+2y)=16 at the…

Q: Prove that the following curves are intersecting with a right angle: 5y-2x+ y -x²y =0 2y+5x+ x* –…

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Q: Use the comparison theorem to investigate whether the spiral r=1/θ starting at θ= 2π and going on…

A: Given spiral is r=1θ. We have to use comparison theorem to state whether the given spiral has finite…

Q: Verify that the following pairs of curves meet orthogonally x = 1 - y2, x = 1/3y2

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Q: Compute the length of the curve r(t)= (-5t, 4t – 3, – 5t + 5) over the interval 0 <t< 7

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

Q: Given that the line y =r- b is an asymptote to the ar curve y a,b €R,r = 1, then b = I-1 -1-a B 1-a…

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Q: Let C be the curve r(t) = (2t, t², t°/3). (a) Compute the Frenet frame for C. (b) Calculate the…

A: The position vector is rt=2t,  t2, t33. Differentiate with t and obtain the unit tangent vector:…

Q: Consider the piecewise smooth curve C = C1 u C2 u C3 u C4, oriented from the point A(1, 1, 2) to…

A: Given that, F=4x3-2z, 3y2,-2x Find the value of ∫cF. dr we have to find the function f such that…

Q: The necessary and sufficient condition for a given curve to be a plane curve is that t = O at all…

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Q: Let C be the curve y = x³ – 3x2 + 4x – 1 joining the points (1, 1) and (2, 3). Find the value of…

A: We need to find value of integral.

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that F.…

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Q: Show that the condition that the curves ..(1) ..(ii) ax? + by =1 %3D a'x² + b'y² = 1 1 1 1 Should…

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Q: Verify Green's Theorem for: f4y³dx-2x²dy where C, oriented anti-clockwise, is composed of the…

A: To find the integral ∮C4y3dx-2x2dy where C is the positively oriented closed curve x2+y2=9, y≥0 and…

Q: Let C be the curve connecting (0,0,

A: Given that C be the curve connecting  (0, 0, 0) to (1, 4, 1) to (3, 6, 2) to (2, 2, 1) to (0, 0, 0).…

Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

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Q: Let C be the curve y = 2/r for 1.3 <x < 3.2.

A: Given that: y=2x

Q: Consider the path r(t) (14t, 7t, 7 In t) defined for t > 0. Find the length of the curve between the…

A: Given - r(t) = ( 14t, 7t2, 7lnt )  defined for t&gt;0 To find - The length of the curve between the…

Q: Let c be a smooth curve defined by r(t)= with 1

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Q: Find the exact length of the loop of the curve = 3t – t°, y = 3t2 -

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Q: 4. Prove that the following curves are intersecting with a right angle: 5y-2x+y²-x²y = 0 2y +…

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Q: the compler pleme Sketcl given by the Curve in Iz-31=12+3il %3D

A: Just use definition of modulus of a complex number.

Q: Compute the length of the curve r r(t) = (2t, In t, t²) over the interval 1 < t < 3 L =

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

Q: Consider Gollow Sx2 of Eunluate y when y to Find y Evalnate of when 11 Write Mhe etios to the Write…

A: Now when x = 1, Now differentiating given equation with respect to x, we get Now when x = 1,

Q: Consider the intersection curve C in the first octant, between the surfaces s1: 2 = 1- r? S2 : 2 +…

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Q: ind the orthogonal trajectory of the curve y? = ax²(1- cx); with a held fixed.

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Q: Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D…

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Q: You can see the question in the picture

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Q: apply stok's Theorem to evaluate (Ydz +7dy+ xdz) whereCiB the curve of intersection of x²+ y²+z² =…

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Q: Sketch the plane curve r(t) = t2i + 2tk, and find its length over the given interval [0, 3]

A: We have, r(t)=t2i+2tk  we have to determine the length of the curve over the interval [0,3]

Q: Sketch the reglon enclosed by the glven curves. x = 6y2, x = 4 + 5y2 y y 2- 2- 10 15 20 10 15 20 -1…

A: Here we have,  x=6y2 ............................................. (1) and, x=4+5y2…

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.