Evaluate F. dr for the vector field F(x,y) = xy²i+ x²yj where C is given by r(t) = cos (t)i + 2 sin (t)j for 0

Q: Given f(x, y)= y sin(xy). find the gradient vector field くFx, Fy> =くy^2cos(xy), sin(xy)+xycos(xy)

A: We have to solve the gradient

Q: 5. Consider the vector field F(x,y,z) = yzi + xzj+(xy+2z) k and the curve C given by r(t) = (t+ sin…

A: Given vector field: Fx,y,z=yzi+xzj+xy+2zk Curve C is given by: rt=t+sin12πt,t+cos12πt,2t,   0≤t≤1…

Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

Q: Consider the vector field F (2x + 5y, 5x + 7y) Is this vector field Conservative? Conservative If…

A:

Q: 1) Consider the vector field F(x, y,z)= 2xyi+(x² +z² )j+2 yz k . a) Find div(F(0,2,3)) b) Find curl…

A: topic - divergence and curl of a vector a) 8  b) 0

Q: Find the divergence of the vector field.

A: To Find the divergence of the vector field.

Q: Evaluate the line integral F dr of the vector field F (x, y) = (x - y)i + (x + y) j C; r (1) = 4 cos…

A:

Q: Consider the vector field F = (5x + 6y, 6x + 4y). F - dr along the curve C given by r (t) = (t², t*)…

A: ∫CF→·dr→=∫abF(r(t)) r'(t) dr

Q: Find an expression for the vector field F = ⟨x - y, y - x⟩ (in terms of t) along the unit circle…

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

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

A:

Q: For the following practice question, please determine whether the vector field is conservative. If…

A: We have to find potential function

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

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Q: Consider the vector field F = (7x + 3y, 3x + 2y) Is this vector field Conservative? ( Conservative…

A:

Q: Determine whether the following vector field is conservative on R. F= (e 2X cos (-2y), e 2 sin (-2y)…

A: The objective is to determine whether the given vector field is conservative or not

Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

Q: Consider the following vector field. F(x, y, 2) = (4e* sin(y), 9e' sin(2), 8e" sin(x) (a) Find the…

A: Solve the following

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: Let a vector field F(х,у) 3sinx 1 + ху ј be given. What is the integral representing the work done…

A: Workdone by force along the path is integration (f.dl) Where f is force and dl is distance vector…

Q: Is the vector field F = y? – 2x², 2xy + 3| conserva Yes No yz Is the vector field F y? – x2 con Vy?…

A: According to the guidelines issued by the company we are supposed to answer only first question.…

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

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Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

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Q: (b) For the case a = 3, b = 4, and for the constant vector field F(x,y,z) = 4 1 + k , find the flux…

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Q: Consider the following vector field F (x, y, z) = yz cos (xy) i+ xz cos (xy) j+ sin (xy)k (a) Use…

A:

Q: Is the vector field F(x, y) = (sin(x)ely=3 + 2r cos(r>)y°, -2e)y-3 + 6 sin(x³)»*) conservative?…

A: Consider the given vector field.

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: Find a potential function for the conservative vector field. F(x, y) = xex*4Y(2yi + xj)

A: If curl f is zero then conservative field

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 <x2y3,x3y2>. 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: Given the following vector field and oriented curve C, evaluate |F•T ds. F= (x,y) on the parabola…

A:

Q: Let F be a vector field and Ca curve with parametrization 7 (t) where a <t<b. F. dr = | (cos(t³),…

A:

Q: Given the vector field F (x, y, z) = 2xyi + (x² + z?) j +2yzk a. Verify that the vector field…

A: Ans :A)

Q: Is the vector field F(x, y) = (sin(x) cos(y), cos(x) sin(y)) conservative? Select the correct answer…

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Q: Consider the vector field F = (4x + 5y, 5x + 7y) Is this vector field Conservative? Conservative If…

A:

Q: Calculate the work done by the field F when the object moves along the given path from the initial…

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Q: Consider the vector field F = (x*y, x*y? Is this vector field conservative? Select an answer If so:…

A:

Q: 3. Use the divergence of the vector field to compute F(x, y) = (tan-1 (2x)) ₁ + (xy-2 +²8²) J j 4 fo…

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Q: Let D be the upper half-plane (y > 0) and consider the vector field F : D+R² given by F(x, y) = (…

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Q: Consider the vector field F and the curve C below. F(x, y) = x2y³i + x³y?j, C: r(t) = (t2 – 2t, t2 +…

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

A:

Q: Use the divergence of the vector field to compute F(x, y) У · (tan-¹ (2x))i + (xy − 2 + 8x²) j - = 4…

A: we are given the vector field as  F(x,y)=tan-1(2x)4i+(xy-y2+8x2)j to compute…

Q: Find the divergence of the vector field V(x, y, z) = -2xi + (y + 5 cos(x))j+ (7z + e8xy) k.

A:

Q: Consider the one-form w = x' cos(y) dx + xe13% dy+4z² dz on R3. The associated vector field is: F(x,…

A: For the solution follow the next step.

Q: Evaluate ∫C F . T ds for the vector field F = x2 i - y j along the curve x = y2 from (4, 2) to (1,…

A: Given information: The value of F=x2i+yj, x=y2 and from (4, 2) to (1, -1). Calculation: Consider…

Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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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: