Q1. Verify Stokes' theorem if F is the vector field as below function and S is the portion of the paraboloid z=3+x²+y? bounded by the planes z=3 and z=7. Take n to be the outward unit normal vector to S. F = 4yi – xj+ 22° k %3D Verified Not Verified - 24n - 267 - 28 - 30n

Q: Find a non-zero vector 7 perpendicular to the vector u 12 [3]. -6

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Q: Given the vector field and oriented curve C, evaluate F.Tds. C (x.y) on the line segment r(t) = (t,…

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Q: What is the flux through the described surface?

A: The surface is described by The vector field is F=<1, 3x, 2y>To evaluate the Flux for the…

Q: X=r sinø cose y=r sinø sine z=r cosø p² = x² + y² + z² Determine a, y ,əz , ə, ,dg,dg

A: Given that: x=rsinϕcosθy=rsinϕsinθz=rcosϕr2=x2+y2+z2 In partial derivatives, except for the variable…

Q: (6.) Let u (1,3,-2) and v (0, 2, 2). %3D (а. Determine compyu the scalar projection ofu onto v. (b.…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 1. For each curve, construct dr. • Let C1 be r? + y? = 16 • Let C2 be y = 5 – 2x • Let C3 be r = 2 –…

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Q: Compute the curl of the following vector field. F= (3z sin y, 3xz cos y, 6xz sin y) The curl of F is…

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Q: 41.) Use given grid to sketch the vector field F = (y, -x). Include vectors at all integers points…

A: Given vector field is F=12y,-x

Q: -1 15. Determine if span R2. 1

A: A set of vector S is said to be span Rn if every vector of Rn can be written as the linear…

Q: Let F be the vector field (ze"+yz², xz², xe"²+2xyz) Let C be the path r(t) = (sin(t), teʻ, cos(t)),…

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Q: Consider the following function. T: R² - R?, T(X, V) = (Vã, sxy, Vỹ) Find the following images for…

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Q: Positive Positive Negative WIN WIN AS

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Q: In the ring Z find the product (mZ) (nZ) and intersection (mZ)n (nZ) where m, n € Z. When are the…

A: (.)   ℤ is a ring of integers . And ideals of ℤ are of the form  nℤ  ;  n∈ℤ . (.)  Let  R be a ring…

Q: A vector field is specified as G = 24xyax + 12(x² + 2)ay + 18z²az. Given two points, P(1, 2, - 1)…

A: Given: G=24xyax+12x2+2ay+18z2az P1,2,-1 Q-2,1,3 To find: c Unit vector directed from Q to P. d…

Q: 2s Let H be the set of all vectors of the form Find a vector v in R' such that H span {i}. -s S

A: Spanning set of H

Q: Let F be the vector field (-2 sin(2x - y), sin(2x - y)). Find two non-closed curves C₁ and C₂ such…

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Q: Suppose C is the closed, oriented curve in the figure. Sketch each vector field F given below and…

A: The vector fields are as follows:To Find:Determine whether the above vector field is positive,…

Q: Consider the vector field shown in the figure below together with the paths C₁, C₂, and C3. ? C3 <?…

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Q: 2. Let F(x, y) -yi-zj + zk be a vector field. Compute F FdS, where S is the hemisphere a² + y² +₂²…

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Q: Consider the following function. T: R? - R, T(x, y) = (2x?, 3xy, y²) Find the following images for…

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Q: Given the vector field F, (a) determine the vector ₹ evaluated at P (r = 2,0 = 2,0 =,=); (b)…

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Q: V = 6y – 16x + 9z + V¢ Find the curl of vector V in Cartesian coordinates, o is Continuously…

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Q: Let F = SOFF Use Stokes' Theorem to evaluate . dr, where C is the triangle with vertices (6,0,0),…

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Q: Let C be the straight line segment in the xy-plane from the point (1, 2) to the point (5, 3). Let F…

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Q: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

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Q: Find the k-component of (curl F) for the following vector field on the plane. F = (x + y)i + (8xy)j…

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Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: