3. Evaluate |F. dr where C is the curve F(t) = (t, t°, t“), 0

Q: If C is the curve given by r (t) = (1+2 sint)i+ (1+ 2 sin? t)j+ (1+1 sin' t) k, 0<t< and F is the…

A: Given:

Q: Find the divergence of the vector field F = < yx®, xy®

A: Given,         The vector field is F¯=&lt;yx3 , xy6&gt; .

Q: Evaluate F. dr for the vector field F(x,y) = xy²i+ x²yj where C is given by r(t) = cos (t)i + 2 sin…

A:

Q: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

A:

Q: Q3 Given the conservative vector field, F =(x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

A:

Q: 2. Find the curl of the vector field. F = z°e* sin(2 y)i – e' cos yj+ xyzk to

A:

Q: Q3 Given the conservative vector field, F = (x +y²)i +(2xy +5)j , evaluate the integral F•ds where C…

A:

Q: Consider the vector field F(x, y, z) = (2ry, x², 1). (c) Calculate the line integral / F - dĩ where…

A:

Q: (a)the curl and (b)the divergence of the vector field. F(x,y,z)= ln(2y+3z)i+ln(x+3z)j+ln(x+2y)k

A:

Q: Siven the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

A:

Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x,y) F = on the curve r(t)…

A:

Q: Find the work done by the vector field F = (– 4z, – 3y, – 2x) in moving an object along C in the…

A:

Q: 2.) Calculate the curve integral over the vector field V(7, y) = ( -x*y xy for the positive oriented…

A: Let x=2cos t y=2 sin t dx=-2sint dt dy=2 cost dt

Q: 2.) Calculate the curve integral over the vector field V(17, y) = ( * -2²y xy? for the positive…

A: Disclaimer: Since you have asked for a handwritten solution and currently we are not providing any…

Q: 3. Consider the vector field F = 2xy i + (x² + 2y) j. Verify the Green's Theorem in the closed…

A: Definition of green's theorem :

Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

A:

Q: b) Find the flux of the vector field F(x,y,z) = 3xi – yj + 2zk; across o where o is the portion of…

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: Find the curl and divergence of the vector field. F(x,y,z)=x^2yz i -x x^3y k

A:

Q: Consider the force field F = (ev – e2" , (x + })ev) and the curve C = C¡U C2 where Cı is the line…

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

Q: Let W be the work done by the forced field F(x,y,z) =yi+zj in order to move a particle along the…

A: The formula for work done is W=∫F.dr.

Q: (c) Determine the flux of the vector field F = x³î – 2yĵ + 7z²k across a cone z = Jx2 + y2 + 3 that…

A:

Q: Find the work done by the field F(x,y, z) = (6z, 12x, y?) along the curve r(t) (cos(t), sin(t),…

A:

Q: Find the work done by the vector field F = ( – 2, – 4y, 2æ) in moving an object along C in the…

A: Let F be a vector field defined on a smooth curve C whose parametric equation is given by rt, a≤t≤b.…

Q: Let ryz , (교2 + y"), 17 F = ( x2 +y? be a vector field on U = {(x, y, z) E R³ | (x, y) # (0, 0)}.…

A:

Q: 3. Consider the vector field F(x, y) = (2xy, x² – sin y). Evaluate %3D F. ds where C is the curve…

A:

Q: Consider the vector field F = (r²y³, x°y?) Is this vector field conservative? Conservative If so:…

A: Given the vector field &lt;x2y3,x3y2&gt;. We need to: i) Check whether the vector field is…

Q: Integrate the vector field h(x,y) = y^2i + (2xy-e^2y)j over the circular arc C parametrized by r(t)…

A: The answer is given by using the formula of line integral

Q: Ex: Find the work done by the vector field F = (y-x²)i + (z - y²)j + (x-z²2)k Over the curve r(t) =…

A: Given: F=y-x2i+z-y2j+x-z2krt=ti+t2j+t3k

Q: Compute the work done by the force field F = , 0 <t<1

A:

Q: Compute the line integral of the vector field F=⟨6y,−6x⟩ over the circle x2+y2=1 oriented clockwise…

A: Given: F=6y dx - 6x dyand Let  x= cos t,   y= sin t    ⇒   dx = -sin…

Q: Find all potential functions of the conservative vector field F (x, y, z) = (2xz + y², 2xy,x² + %3D…

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Q: Find the divergence of the vector field F. F(x, Y, z) = In( 9x²+ 8y² )i + 72xyj + In(8y² + 7z? )«…

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Q: 2.) Calculate the curve integral over the vector field V(7, y) = ( -x*y xy² for the positive…

A: Green's theorem

Q: Find the work done by the force field F(x, y) = xi + 2yj on a particle moving along the path C:x =…

A:

Q: - L ---- Let F be the vector field (r², 2r, z²). Let C be the boundary of the surface S given by 4x²…

A: Given- Let F→ be the vector field x2, 2x, z2. Let C be the boundary of the surface S given by 4x2 +…

Q: Find the work done by the vector field F = (- 3z, 4y, 2x) in moving an object along C in the…

A: Work done by any vector field F is given by ∫F.dr where r is the position vector. Dot product is the…

Q: Given the following vector field and orientated curve C, evaluate F•Tds. (x.y) F = on the curve r(t)…

A:

Q: Let B(t)-(t,t) be a smooth curve in R. Deter min e a vector field Y =(y- 42) € X(R) with Ya) =(1, 2)…

A:

Q: 2) Determine the work done by the vector field F along the curve y. F(x, y. =) = (x sin y, 2e*, 2xz)…

A: First I have written about the work done by the given vector field in the direction of the given…

Q: Consider the line integral a xy dx + yz dy + xz dz. (a) What is the vector field that is being…

A:

Q: Let F = Vx, 0, -/z] be a vector field and C be the curve given by the parametrization r(t)=…

A:

Q: 16. Compute the flux V· nds of the vector field V(r, y) = (x² – 2ry, y² – 2xy) across the curve C :…

A: We have to compute the flux ∫ℂV·nds of the vector field Vx,y=x2-2xy,y2-2xy across the curve…

Q: For the force field F = (2xe¬Y, (2y – x²e¬y)) in some open connected region containing P(1, 0) and…

A: Given that, F=2xe-y, 2y-x2e-y If F is conservative then ∂2F∂x∂y-∂2F∂y∂x=0

Q: 21. Compute the flow of the vector field F = r through the surface of the ellipsoid y² + 2 = 1 in…

A:

Q: 3. For the vector Field E = -(x² + 2y²)âx + (xy)ŷ calculate E dl around the triangular contour (V×…

A:

Q: Find the work done by the force field F(x, y) = xyi+ x²j on a ,2 particle that moves along the curve…

A: Consider the given force field Fx,y=xyi+x2j Then, F⇀x,y=xy,y Write in parametric form y=t  then…

Q: Consider the vector Field 7 and the curve C below. F(x,y) = (3 +4 ху?) 1 + 4x²yј, C is the arc of…

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Q: Find the divergence of the vector field F. F(x, y, 2) = In(7x2+ 6y? )i + 42xyj + In(6y2. + 72² )«…

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Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

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: