Show that F. dr is independent of its path, given that F(л, у, 2) — + tan 1+x² 2) i+ tanrj+ k. 1+2² Hence, find the potential function of F and then evaluate the work done by F moving an object along the smooth curve C, the line segment from (1,0, – 1) to (0,0, 0) then along the parabola z = y°, x = 0 to (0, –1, 1).

Q: Show that / F dr is independent of its path, given that F(r, y, :) = + tan-:) i+tan rj+ k. IHence,…

A: Here we will find out the required value and prove the required result.

Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

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Q: b) Show that F - dr is independent of its path, given that y F(r, y, 2) = 1+. + tan-2) i+ tanrj+ k.…

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Q: F(x, y) = (8xy° + 3)i+ (12x²y² + 2e2")j is conservative by finding a potential function f for F, and…

A: Given that, Fx,y=8xy3+3i+12x2y2+2e2yj⋯⋯(1) C: rt=2sin5ti+2tπsin85tj 0≤t≤π2S Let, F=Mi+Nj Where,…

Q: Need correct solution ASAP

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Q: F(x, y) = (8xy + 3)i + (12x²y² + 2e²")j is conservative by finding a potential function f for F, and…

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Q: Without using a potential function, find the total work performed by the force field F(x, y) = = (2x…

A: Given the vector field F→x,y=2xsiny+y2ex,x2cosy+2yex. The particle moves along the curve x=sinπy2…

Q: 6: Find the work done by the force F = -7y i+ 10x j along one loop of the curver = sin 70 (in the…

A: Given

Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

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Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

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Q: Show F(x, y) = (2xy³ + 8)i + (3æ²y² + 2e2")j is conservative by finding a potential function f for…

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Q: Evaluate the line integral F. dr where F = (3 sin x, 3 cos y, 10xz) and C is the path given by r(t)…

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Q: Let flxiy) = Rx Jy 27 %3D a) Finid the gradrent of f b) Find the directional derivative of f at the…

A: Consider the function as  fx,y=2xy

Q: Evaluate the line integral F. dr where F = (3 sin x, 3 cos y, 10xz) and C is the path given by r(t)…

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Q: What is the correct piecewise parametrization of a path C created by starting from C₁ of equation y…

A: Introduction: A very important aspect of vector calculus is the parametrization. the line integral…

Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Show that F dr is independent of its path, given that F(r. y. :) = ( i+ tanrj+, + tan k. 1+:2 %3D…

A: Here we will find out the required value and prove the result.

Q: Show that / F. dr is independent of its path, given that F(x, y, 2) = ( : 2) i+ tan-lr j+ k. 1+ 22 +…

A: Here we will find out the required value.

Q: (B) Find the potential function V = V(r, y) such that %3D + yi+ + r)j = V¥ 1+ x2 + y? 1+ x? + y? and…

A: We have to find the potential function ψ=ψx,y such that x1+x2+y2+yi+y1+x2+y2+xj=∇ψ and also evaluate…

Q: Evaluate the line integral f. F. dr where F = (2 sin a,–5 cos y, 10xz) and C is the path given by…

A: We will evaluate the given line integral.

Q: Sketch the direction field of the D.E. y = y - t and the solution curves corresponding to y(0) =-1,…

A: Solution of question is written in below steps.

Q: Find a potential function for F or determine that F is not conservative. (If F is not conservative,…

A: Given Data The function is F=<6zsec²(x), 6z, 6y+6tan(x)>.   The curl of the vector field…

Q: Let F(x, y) = (4ye*y – 3x², 4xe*y – sin y). %3D a. Find a potential function of F. b. Find S„F· dR…

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Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

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Q: Evaluate F. dr where F = (x – 3y)i + (y – 2r)j and C is the closed curve in the xy plane, x = 2 cos…

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Q: Without using a potential function, find the total work performed by the force field F(x, y) =…

A: The work done by the path E is given by, W=∫EF.dr If the force vector is: F(x,y)=<2xsiny+y2ex,…

Q: Consider the curve C in RS parameterized by r(t) = sin(2t) î + cos(t) ĵ+t k, con 0 <t < 2n An…

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Q: Show F(x, y) = (4xy³ + 6)i + (6x²y² + 2e²¹)j LF is conservative by finding a potential function f…

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Q: Let F = (e" sin y) i+ (e cos y + y)j. Answer the following questions. (a) show that is F…

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Q: Show that the line integral is independent of path tan y dr + r sec y dy where C is any path from…

A: Given that ∫C tany dx +x sec2y dyShow that the line integral is independent of path and evaluate the…

Q: Use Green's Theorem to evaluate the line integra Icon (y − x)dx + (2x − y)dy - - for the given path.…

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Q: 5. Compute the lenght of the curve u = t, v=0 from t = 0 to t = 1 using the first fundamental form.

A: Suppose, a continuous function is provided over a certain interval. The arc length is the length…

Q: 2. Find the gradient field F = Vy for the potential function p. Sketch a few level curves of o and a…

A: Given φ(x,y)=2x2+4y2 The gradient field  F=∇·φ=∂φ∂x,∂φ∂y=4x,8y Some vectors of F are  At x=1 and y=1…

Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

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Q: Evaluate the line integral F. dr where F = (-3 sin x, cos y, 5xz) and C is the path given by r(t) =…

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Q: B) Evaluate the line integral F-dr, where F (x, y) = x²y³i+ x°y°j and C is the curve r (t) = (t³ –…

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Q: Use Green's Theorem to evaluate / F · dĩ, where F = (VE + 3y, 2x + vũ) C consists of the arc of the…

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Q: Let C = C₁ U C₂ U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

A: The main objective is to show that F is conservative vector field. Find the potential function. Also…

Q: Does f(z)=z/(sin z)^2 have a pole of order 1 or 2 at z=0?

A: f(z)=zsin2zFor the poles first we look at the zeros of the denominatorat z=0 We know sin(z) has a…

Q: Without using a potential function, find the total work performed by the force field F(x,y) = (2xsin…

A:

Q: F(x, y) = (8xy' + 8)i + (12x²y² + 2e²)j is conservative by finding a potential function f for F, and…

A: We have used the conservative vector field formula since F is conservative vector field then we…

Q: Without using a potential function, find the total work performed by the force field F(x, y) = (2x…

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Q: Let C denote the curve of intersection between x = -5z° and y = 6z", starting from A(0,0,0) and…

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Q: Without using a potential function, find the total work performed by the force field F(x, y) =…

A: Work done by vector field while moving a particle along a curve = Line integral of that vector field…

Q: Calculate the line integral in scalar field ∫C (x2y2-√x)dy is the arc of the curve y=√x of (1,1) to…

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Q: 7. Let F = (2x sin(y)e", x² cos(y)e*, z² sin(y)e*). Find along the path c:T = (sin(t), t, cos(t))…

A: F=2xsinyez, x2cosyez, x2sinyez curlF=ijk∂∂x∂∂y∂∂z2xysinyezx2cosyezx2sinyez⇒curlF=0 So, F is a…

Q: Given that F(x, y, z)=sin x i+cos yj+ xzk. Evaluate , F.dr where C is given by r(t)={r',-f,t) :…

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Q: What is the correct piecewise parametrization of a path C created by starting from C₁ of equation y…

A: Concept: We have given the curves  y=x+3x=0 to x=4y=x2 Here the red line represents y=x+3 The green…

Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, -1) to (0, 0), C₂ is the line segment…

A:

Q: Let C = C₁ U C2 U C3, where C₁ is the line segment from (-1, −1) to (0, 0), C₂ is the line segment…

A: a. Suppose,  F=∇fF=∂f∂x,∂f∂y So, ∂f∂x=12x+2+tan-1y∂f∂y=x1+y2+3y2 Now, ∂f∂x=12x+2+tan-1y Integrate…

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.