In Problem 3, find the flux of the given vector field F through S according to the described orientation, where S is the portion of the cylinder of radius 2 centered along the z-axis that lies between the planes z = 0 and z = 3 in the first octant. The surface S is shown in the figure to the right. 3. F = yi+ 2xj+ 3zk, and S is orieted away from the z-axis.

Q: Given the vector field A 4rf + 2.6rsin(0)sin(ø)0 – oo. The result of p0. The result of SA• dl along…

A: The line integral is:

Q: The temperature function on a circular plate R = {(r, y) |r² + y? < 36} is given by T(r, y) = 200 –…

A: Consider the temperature function given by  T(x,y)=200-x2-y2  on a circular plate…

Q: 3. Consider the following helix with vector equation 7 (t) = 2 cos (t) 7 (t) 7 + tỉ + 2 sin (a)…

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Q: three decimal pI 1.1 ,Yi = i 2.51 X2 = i 1.2 ,Y2 = i 3.273 X3 = i 1.3 Y3 = 4.50283 X4 = i 1.4 Y4 =…

A: Consider the equation

Q: If F(x,y,z)=1i+2j+3k, then O1. P=1 Q=2 and R=3 are vectors of F. O II, P=1 Q=2 and R=3 are gradient…

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Q: Stoke's theorem

A: We have to verify the Stoke's theorem for the given vector field. It is given that

Q: Problem #9: Find the flux of the vector field F = xi + ex j + k through the surface S given by that…

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Q: A sodium ion (Na+) moves in the xy-plane with a speed of 2.90 ✕ 103 m/s. If a constant magnetic…

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Q: Is the two-dimensional vector field V = -2y 2x a possible velocity field x²+y² 'x² +y² for an…

A: For a two dimensional incompressible for a vector field V=ui+vj. ∂u∂x+∂v∂y=0

Q: - Suppose the vector field v(x, y, z) = (2z − 3) i+3zj + (3y − 1) k represents the velocity field of…

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Q: 10. Given F(x, y) = (x, y), find the flux of the vector field across the piece of the circle x² + y²…

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Q: A particle is moving along the curve having the vector equation R(t) = ti+t²j+ tk. Find the vectors…

A: The given position vector is  Rt=t i + t2 j + t k.   The velocity vector is given by Vt=R't=i+2t j…

Q: 4. Find the work done by the force field F = (x? sin(y),y) on a particle that moves along the cubic…

A: Given: The force field, F→=x2siny,y on a particle that moves along the cubic y=x3 from 1,1 to 2,8.

Q: 2. DS: phase-plane, equilibria a. Identify each phase portrait below with a vector field from the…

A: Given F=(y,-sin x) G=1-x-y/2x, 1-y-2x/3y H=y,x-x3

Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

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Q: 5. (a) Use Stokes' theorem to calculate the circulation of the vector field F around the curve C.…

A: Stokes theorem:

Q: The vector field V(r) = v(p) where p = √² + y², and is the unit vector of polar coordinate. √/z²+y²…

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Q: 3) Find the curl of the vector field F(r, y, z) = (ry") i + (ya)j+ (8ry) k %3D

A: We know,If, F(x,y,z)=Fx i +Fy j+Fz kCurl of  F(x,y,z)=∇× F(x,y,z)=ijk∂∂x∂∂y∂∂zFxFyFz

Q: 3. A particle moves in the xy-plane so that at any time t, its coordinates are given by x = t° -1…

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Q: match each of the following planar vector fields with the corresponding plot in Figure 10. F = (x +…

A: To match planar vector field with given figureWhere F=(x+y,x-y)

Q: 1. The acceleration of an object is a=2i+4j+6k.  Compute the position vector r(t) if the object at…

A: a=double derivative of position vector

Q: If vector T = (x+ y+1)i+ j– (x +y)k then T.curl(T )is

A: We are given vector T.

Q: (0, x,0). Plot this vector field and find its Suppose the velocity field u is given by u = flow…

A: Let for convenience, unit vectors along xi axes be ei. and Di=ddxi So given velocity field u=xe2 So,…

Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

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Q: 1. Evaluate 3 √² 13 0 |3x² - 12 dr.

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Q: Find the flux of the vector field F = 2 i +3 j – 2 k through the rectangle with corners at (1,0,0),…

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Q: y? = 1 and the vector field F = xy?i+ 2x²y}. (c) The ellipse %3D 4

A: Solution c: Given  Ellipse's equation x29+y24=1 and vector field F→=xy2i^+2x2yj^ Parametric equation…

Q: 1. Find all the potential functions, if there are any, of the vector field F(x, y, z) = (2x sin(y −…

A: Given vector function is                  F→x,y,z…

Q: 3. Consider the vector field F(x, y) = (3y²+ 4x, 5y sin(x)). (a) Calculate curl F at the point (0,…

A: The curl of a vector field F = (P, Q) in two dimensions is given by the formula:∇×F=∂Q/∂x-∂P/∂yHere,…

Q: Use Green's Theorem to find the flux of the vector field F(x, y) = (x + y, −x² − y²), where is the…

A:

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: