Which of the following statements are true for all vector fields, and which are true only for conservative vector fields? (a) The line integral along a path from P to Q does not depend on which path is chosen. (b) The line integral over an oriented curve C does not depend on how C is parametrized. (c) The line integral around a closed curve is zero. (d) The line integral changes sign if the orientation is reversed. (e ) The line integral is equal to the difference of a potential function at the two endpoints. (f) The line integral is equal to the integral of the tangential component along the curve. (g) The cross partial derivatives of the components are equal. 3

Q: 2. Show that for an arbitrary scalar field ø and vector field a (i) V×(Vø)=0 (ii) V.(V×a)=0. (iii)…

A: In vector calculus these derivations have lot of importance. They can be proved very easily and…

Q: 9) Check whether the vector field F = (-cos x cos y, sin x sin y, - sec² z) is a gradient vector…

A:

Q: 4. Consider the vector field given in Cartesian coordinates by A(r) = (x² + y³, 3xy², z²). (a)…

A: Given : Ar=x2+y2,3xy2,z2

Q: Given a two-dimensional vector field F and a smooth orientedcurve C, what is the meaning of the flux…

A:

Q: Determine whether the vector field F(x, y) = 5y² (yi+2xj) is conservative. F is not conservative F…

A:

Q: Is the following vector field conservative? F =

A: Yes the vector field is conservative

Q: Consider the vector field F that is depicted below. On which of the parameterized curves is the…

A: Given the vector field F in the image

Q: Let the vector field F (x, y) = (3x2y-2+ 2xy-1)i + (-2x3y-3 − x2y-2)j, be a conservative field.…

A: We will use Integration and concept of total differentiation for finding scalar potential of vector…

Q: Show that the vector field F = (x² cos y, e+y) is not a gradient vector field. %3D

A:

Q: Show that Gauss's Theorem holds for the vector field F(x, y)=(xy2, -yx2) in a side square 3.

A:

Q: Supposing vector field G is related to the flow of fluid P is one point in the domain of G. Give the…

A: To Explain: The geometrical interpretation of G(P) where G is a vector filed and P is a point in…

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

A:

Q: How do you determine whether a vector field in ℝ2 is conservative(has a potential function w such…

A:

Q: Write a formula for a two-dimensional vector field which has all vectors of length 1 and is…

A:

Q: When is a vector field irrotational?

A: We have to determine when is a vector field is irrotational.

Q: ... (a) Is the vector field conservative? Why or why not? (b) Does the curl of the vector field at P…

A:

Q: Explain why a potential function for a conservative vector field is determined up to an additive…

A:

Q: Determine whether the following vector field is conservative on R°. If so, determine the potential…

A:

Q: Find the equations of the field lines for the vector field defined by the two dimensional field, F =…

A: Given vector field F=xi^-yj^....(1) Compare the equation (1) with F=dxdti^+dydtj^, we get  dxdt=x…

Q: Write a formula for a two-dimensional vector field which has all vectors of length 7 and…

A:

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: Is the vector field F (x, y) = (2xy – y³) î + (x² – 3xy²) ĵ solenoidal, conservative, both or…

A:

Q: Provide two examples of physical phenomena in which the divergence of a vector field is nonzero.

A: Q: What is Divergence? A:  It operates on vector field (having magnitude + direction both) , for…

Q: Determine whether F(x, y) = 3xi + 7yj is a conservative vector field. find a potential function for…

A:

Q: Consider the vector field F = ( 'x²+y2 'x²+y2/ Write yes or no for each: Is this field conservative…

A: The solution are next step

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: Show that the vector field F(x, y, z) = (-ycos(8x), 8x sin(-y), 0) is not a gradient vector field by…

A:

Q: Show that the following vector field is conservative and find the potential F(r, y, z) = (2ry, x² +…

A: The answer is given as follows.

Q: Determine whether the following vector field is conservative on R°. If so, determine the potential…

A: If curl f is zero then conservative vector field

Q: determine if the given vector field f is conservative and hence find the potential function for the…

A:

Q: The movement of air on the surface of a certain closed controlled space is studied, and every point…

A:

Q: Consider the vector field F(x, y) = (x², −y). If (x, y) is in the first quadrant then the vector…

A: Part 1. (x, y) is in the first quadrant.  => x > 0 and y > 0 => x2 > 0 and - y < 0…

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

A:

Q: A=(2x-3y)ax+(2xy-y2)ay Prove the Stokes theorem for the triangular linear path in the shape that is…

A: The general form of a vector valued function in a plane is given by A=Axax+Ayay, where Ax and Ay are…

Q: (a) For any vector field F, the gradient of the divergence of F is always zero.

A: Since the gradient of divergence of F is given by

Q: Write a formula for a two-dimensional vector field which has all vectors parallel to the y-axis and…

A:

Q: Determine whether the vector field is conservative. F(x, y) = xe(x^2)3y(2yi + xj)   If it is, find…

A:

Q: 1. Is the vector field V (x, y, z) = (2ry, x² – 2², –2yz) conservative?

A: Since you have asked multiple question,as per our guidelines we are supposed to answer only one…

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

A:

Q: Explain why Green’s Theorem proves that if gx = ƒy,then the vector field B = ⟨ƒ, g⟩ is conservative.

A:

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

A:

Q: Determine whether the following vector field is conservative. If it is , find the potential…

A: F=<cos(z0, 2y-xsin9z)>=cosz i+2yi-∅sinz k i) curl F=ijk∂∂x∂∂y∂∂zcos z 2y-xsinz…

Q: Which of the following functions is a potential function for the conservative vector field F (x, y,…

A:

Q: 3. Show that the vector field F = -C with C> 0, is conservative and find its potential.

A:

Q: Define the following :- i) Vector Field. #) Piecewise Smooth Curve C

A:

Q: Let F(x, y, z) =⟨3x2y+az, x3,3x+ 3z2⟩ be a vector field. For what values of a is F conservative?

A:

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: Determine whether the vector field F(x, y) = 5y2(yi + 2xj) conservative. F is conservative Fis not…

A: Just check if curl of F Is zero or not.

Q: a) What is the flux of the constant vector field F = (a, b, c) through any closed surface? b) Is the…

A: Given That : A constant  vector field F=a,b,c and F=x3,y3,z3 To Find : i) The Flux through of…

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: