Let C denote the boundary of the region R, oriented counterclockwise. Consider the vector field F= 3 a) Use Green's Theorem to write a double integral which computes the circulation (work) of F along the curve C: So F•dr = dx

Q: Given the vector field F(x, y, z) = sin zi + cos yj + zzk, evaluate the line integral JF-dr where…

A: If curl F =0 It implies that F is conservative.

Q: Let f(x, y, z) = xy? z3 +4xev - In(x - z). %3D (a) Compute the gradient vector field for f. (b)…

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Q: ) Use Green's Theorem to evaluate the line integral along the given positively oriented curve: F. dr…

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Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the…

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Q: Explain the integral obtained in Green's theorem of a vector field F(x,y)=-(1y/2)i + (1x/2)j that…

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Q: 3) Given the vector field F = circulation around the triangle formed by the intersection of the…

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Q: 4. Show that the line integral of F = [0, z] around a closed curve in the r- y plane, oriented as in…

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Q: Solve using the method of undetermined coefficients: y" + 3y" – 10y' = 2 + xe* %3D

A: Given: y''' + 3y'' - 10y' = 2 + xex

Q: Let F = xyi + yzj + zæk be a vector field. Use Stokes' Theorem to evaluate f. F . dr where C is the…

A: Given that,F=xyi+yzj+zxkand the given plane is,x+y+z=1

Q: 4. Verify Stokes's Theorem for the vector field F(x, y, z) = ( yz, – xz, z) over the cone z² = x² +…

A: a) Given that the vector field Fx,y,z=yz,-xz,z over the cone  z2=x2+y2 with 0≤z≤2 oriented downward.…

Q: Compute the line integral of F along the oriented smooth curve C shown below. (2,0, 2) (1,4, 3)

A: F =(ysinz)i +(xsinz)j+( xycosz)k dr =(dx)i +(dy)j+(dz)k line integralx-21=y-0-4=z-2-1=t (x,y,z)…

Q: Use the surface integral in Stokes' Theorem to calculate the circulation of the field F around the…

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Q: Find the line integral F.dr of the vector field F=ax] over a curve C defined by a C rectangle in the…

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Q: Let S be the surface in the xy-plane defined by the inequalities |æ|+ |y| < 1, k. Verify the Stokes…

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Q: Use the divergence theorem to calculate the flux of the vector field F(x,y,z)=−3xy−2yzj+4xzk…

A: Given: And the radius of sphere S is 3.To Do: Calculate the Flux using the divergence theorem.

Q: AK5: Let S be the part of the graph of the plane z = 1- x - y/2 over the triangle 0≤ y ≤2-2x and 0 ≤…

A: Stokes theorem: If S is a surface enclosed by a closed curve C, then

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: