Q2/ Find T, N, and k for the space curve r(t) = (3 sin t)i +(3cos t)j + 4tk

Q: Two particles travel along the space curves r,(t) = (t, t²,t³) and r2(t) (1+ 2t, 1 + 6t, 1+ 14t),

A: Given data: The first curve is r1(t)=<t, t2, t3>. The second curve is r2(t)=<1+2t, 1+6t,…

Q: Sketch the space curve and find its length over the given interval r(t) = ⟨8 cos t, 8 sin t, t⟩,…

A: Given Finding the derivative with respect to t:

Q: Find a function y = f(x) that describes the parametric x = -2t + 1, y = 3t², for – o <t < ∞. curve

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

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Q: Use Green's theorem to evaluate , (5xy + x2 + y?) dx + (x²-y)dy where C is a closed curve that is…

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Q: At what points does the curve r(t)=(t,t,14t) intersect the surface z=x2+3y2?

A: Given: rt=t,t,14t, z=x2+3y2 for finding intersection point of curve and surface we put point t,t,14t…

Q: Find equation of the plane tangent to the following surface at the given point. an z = tan(xy):…

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Q: Find T, N, and k for the space curves r(t) = (t3/3)i + (t2/2)j, t &gt; 0

A: Given, the space curve r(t) = (t3/3)i + (t2/2)j, t&gt;0 We have to find the T, N, and K for this…

Q: Find the point on the surface z=x2-y2 at which the tangent plane is parallel to the plane…

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Q: Find the coordinates of the centroid of the curvex = et cos t, y = et sin t, 0 ≤ t ≤ p.

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Q: Find T, N, and k for the space curve, where t>0. r(t) = (8 cos t+ 8t sin t)i + (8 sin t- 8t cos t)j+…

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Q: Find T, N, and k for the space curve, where t> 0. r(t) = ( cos t+t sin t)i + ( sin t-t cos t)j + 4k

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Q: Find T, N, and x for the space curve r(t) =7i+j,t>0. T(t) = (O i+ (O i

A: Formula for T, N and k are defined as  T=vv, where v=drdt N=dTdtdTdt and k=1vdTdt

Q: Find F. dr, where F(x, y) = (xy² + cos x)i+(x²y+sin y)j and C is the ellipse C x2 + 2y2 = 9,…

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Q: Find the points on the surface 2a2 + ly² + 522 = 1 at which the tangent plane is parallel to the…

A: We will use the following method for finding the two required point.

Q: Let C be the space curve defined by Ř(t) = (2cos t,2 sin t, t). O Find the moving trihedral of C at…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: Find T, N, and K for the space curve r(t) = (9 sin t) i + (9 cos t) j+ 12t k. T(t) = (O i+ O i+ (O k

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Q: Find T, N, and k for the space curve r(t) = i+Tj, t> 0. T(t) = (O i+ (Oi

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

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Q: Find the equation for the tangent plane to the surface z = - 9x - 7y at the point (2, 1, - 43). O A.…

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Q: Evaluate F. dī where F = , and C' is the circle of - counterclockwise direction.

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Q: Find T, N, and k for the space curve, where t> 0. r(t) = (2 cos t+ 2t sin t)i + (2 sint- 2t cos t)j…

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Q: Let C be the space curve r(t) = (3 sin(t) – 1,4 sin(t) + 1), 0 < t < . The length of C is

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Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Sketch the plane curve. r(t) = 4ti - tj, [0, 4] y y 4 4 2 10 15 10 15 o-4 o-4 y y 4 X 5 10 15 10 15…

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Q: F(x, y) = 2xyi + (y² – x²)j (x2 + y2)2 and C is any positively oriented simple closed curve that…

A: Given  F(x,y) = 2xyx2+y22i+y2-x2x2+y22j To Evaluate  ∫CF(x,y)dr

Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: Green’s Theorem for line integrals Use either form of Green’sTheorem to evaluate the following line…

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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: a) Evaluate y° ds, where the curve C is the intersection between the surface z and the plane x = 3…

A: Integration problem...

Q: Find the points on the curve x2 + xy + y2 = 1 in the xy-plane that are nearest to and farthest from…

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Q: Find the points on the surface 2a2+ ly? + 522 = 1 at which the tangent plane is parallel to the…

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Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: Find an equation of the plane tangent to the following surface at the given points. z =6 cos (x- y)+…

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Q: Evaluate F-di where F(x, y, z)=1li +(x+yz)j+(xy- Vz 3x+2y+z=6 in the first octant. C is a positively…

A: Here we have to evaluate ∫CF→·dr→=? Where F→(x,y,z)=i^+(x+yz)j^+(xy-z)k^ and C is the boundary of…

Q: Compute e3z dz , Jy (z – 1)2(z+ i) where +y is any simple closed curve oriented counterclockwise…

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Q: Let Ci be the rectangle with vertices (0,0), (2,0), (0,1), and (2,1) oriented counterclockwise, and…

A: Let C₁ be the rectangle with vertices (0,0). (2,0), (0, 1), and (2,1) oriented counterclockwise, and…

Q: Find the equation for the tangent plane to the surface z = x2 at the point P (1,1,1). Select the…

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Q: a) Evaluate ds, where the curve C is the intersection between the surface z = V1- y? and the plane r…

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Q: Evaluate § F · ds, where F = 2x(x+y)i + (x² + xy + y?)j and the positively-oriented curve C encloses…

A: Here we will use greens theorem to find line integral of vector field For a vector field F→= Pi^+Qj^…

Q: Consider the curve r (t) = (t, t³, t®); t>0 The curve intersects the plane (in a three dimensional…

A: Topic- intersection point of plane and curve

Q: 8 Find T, N, and k for the space curve r(t) = Gi+gj, t> 0 8

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Q: Find T, N, and k for the space curve r(t) =i+ i, t>0_

A: In this question, we find the unit tangent and unit normal along with the curvature of given r(t)

Q: Find the curvature K for the space curve r(t) = (/3sint)i + (sint)j+ (2cost)k.

A: Let, γ(t) be a space curve. The curvature of the space curve is given by, κ=γ'(t)×γ"(t)γ'(t)3.

Q: Find the curvature of the space curve. r(t) = (6t sin t+ 6 cos t)i + 6j + (6t cos t-6 sin t)k 1 O A.…

A: Given space curve

Q: if f(x,y) =x2y÷x+y and y(t)=  (x(t),y(t)) is parametrized curve in domain of f with y(0)=(2,-1) and…

A: The first step in solving the problem is to obtain the parametric equation. The initial values can…

Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

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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.