8. If F is a conservative vector field and S is a sphere of radius a enclosing a region D, then v.FdV (a) is positive. (b) is zero. (c) depends on F.

Q: Find the divergence of the field. F = (2x+7y-8z)i + (5x +3y+6z)j + (-6x-y+2z)H div F =

A:

Q: Show that work performed by a constant force field F over any path C ~ from P to Q is equal to F •…

A:

Q: 1/2 5) Consider vector field F(x, y,z) = ( 2/yz – 2xz', - 2x²z) for 1/2 x>0, y > 0 and z>0 a)…

A:

Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: 4. Is this vector field F(x,y)=(ycos x)ĩ +(sin x+ y)j being the path-independent field? Please show.

A: To check if the given vector field is path-independent field or not. Given:

Q: (1) Is the vector field F(x, y) = (2æ*y^ + æ) î + (2a*y³ + y) Î conservative? Support your answer…

A: Note: As per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: Problem VI: Define the divergence and the curl of a vector field, and prove, for a vector field F,…

A: we have to define here the divergence and the curl of vector filed F.

Q: In problems 11 and 12, do the computation two ways: i) by using the definition of either flux of a…

A: Using i) the definition of the flux across the surface is  ∬SF.dS.

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

A:

Q: Is the following vector field conservative? F =

A: Yes the vector field is conservative

Q: 7. Determine if the given vector field F is conservative on some domain and, if it is, find the…

A:

Q: 4. (a) Show that the divergence theorem holds for the vector field F = 2xî + yz} – 3z²k where o is…

A:

Q: 8. Given the vector field G(x, y, z) = e>+2z (1 + x j + 2xk) : a) Show G(x, y, z) is conservative…

A:

Q: 5. Compute div F and curl F if F = (2x – cos y)i – x²e32j + (z² – 7x)k. 6. Determine if the vector…

A:

Q: 10. () The gravitational field F(x) = mMG Ex is a conservative vector field.

A: We shall show that F is conservative vector field by using vector calculus please follow next step…

Q: Find the divergence of the field. F= (4x + 5y + 7z)i + (8x - 8y + 3z)j + (2x - 7y + 4z)k div F =

A:

Q: (2) For a differentiable vector field f, which of the following is (are) incorrect? (a) ▼ × (▼ × ƒ)…

A: Since we know Formula a→×(b→×c→)=(a→·c→)b→-(a→·b→)c→

Q: For a force field F, if curl F= 0, then the field is nonconservative field. True False

A: False

Q: 3. Use the fact that F = is a conservative vector field to evaluate SF. dr when C is the path r(t)…

A:

Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: 5. Suppose that the vector field F = is conservative. Find a potential function of the vector field…

A:

Q: 3 2) Integrarte the vector field hlxry,2)= xyi + ya J+xa k over the twisted cubic PIt)=ti+t?g + t3 k…

A: Given that, the vector field        h(x, y, z) = xy i + yz j + xz k        r(t) = t i + t2 j + t3 k…

Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

Q: What does it mean to say that a vector field is conservative?

A: What does it mean to say that a vector field is conservative?

Q: 1. Determine whether F(x, y) et sin yi + e" cos syj is a conservative vector field.

A:

Q: Find div ccurl F) = v(♡XFJ For the V cetor Field given By F(AoyoZ)= X=i -6xzj + yz K

A:

Q: 2. Which of the following statements are true for all vector fields, and which are true only for…

A: according to our guidelines we can answer only three subparts and rest can be reposted.

Q: Find the work done by the three-dimensional inverse-square field F(r) : 1 r on a particle that moves…

A:

Q: 2. Letr =√x² + y² + z². (a) For n ≥ 1 and r> 0, express V(r-n) in terms of r and the radial vector…

A: Given  solution(a): Radial vector field  solution(b):

Q: 3. Determine if F = x*yi+ xyz j + x²y² k is a conservative vector field.

A:

Q: 1. Show that the field F = (x + cos a)i+ (K +4 – y sin y)j + (2³ + K – 9)k is conservative and find…

A:

Q: Consider the following vector field: F = (y* + 1, 3ry + 1). (8.1) Is ſ. F · dr path-independent?…

A: The given vector field is F=y3+1, 3xy2+1. (8.1) For a vector field F=P,Q, the line integral ∫CF·dr…

Q: If F is a conservative vector field on a region R, how do you evaluate∫C F ⋅ dr, where C is a path…

A:

Q: 15. To determine whether a continuous vector field is conservative or not, A. its corresponding…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Find the divergence of the field. F = (- 6x + 8y – 4z)i + (- 8x – 5y + 3z)j+ (x + 5y + 2z)k div F =

A:

Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

A:

Q: If. E-Pi+Q+Rk is.a.veeter field.and. Po.Q.and R. have.centineons. 2nd order.partial.derivatives.then…

A:

Q: 2. Show that F is a conservative vector field and then use that fact to evaluate SF. dr along the…

A: We need to show F is conservative vector field.

Q: 2 1 Given the vector field F=rcos a.+ =sing a tra evaluate the curl of the vector field.

A:

Q: a) What is a conservative vector field?(b) What is a potential function?

A: Conservative vector field: A conservative vector field is a vector field that is the gradient of…

Q: Determine if the vector field F = y ln zi – x ln zj + xyk is a conservative vector field.

A:

Q: 7. Choose an option in which the vector field is conservative: None of them OF = (xy, x+ y) O 2,…

A: A vector field F=∇f is conservative if and only if ∂M∂y=∂N∂x

Q: Find the divergence of the vector field F div F II

A:

Q: i) Show that F is a conservative vector field.

A:

Q: F(x, y, 2) = (y² cos r + 2*)i+ (2y sin æ – 4)j+ (3xz² + 2)k Cos z

A:

Q: EXAMPLE 4 Line integral of a conservative vector field Evaluate |(2ry – z) i + (x² + 2z) j + (2y –…

A: The line integral of a conservative vector field is  C is a simple curve from A(-3,-2,-1) to…

Q: 2. State whether the divergence of the vector field shown below is on average positive, negative or…

A: If the divergence is positive the rate of change is outward like a heated gas that is expanding. If…

Q: (a) Suppose that you are asked to determine the path that requires the least work for a force field…

A: For Conservative Force Field  amount of work it takes to move a particle from one point to another…

Q: Determine whether the following vector field is conservative on R. F- (x.4y)

A:

Q: ermine whether the vector field F(x, y, z) = (x³, 3zy², y³) is conservative on R³. If so, find a…

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: