Exercise * F (x₁y₁ z) = (ay³ - 2 z)i + (2xy² + 4yz)j + (bx - cy²) K a) find a, b and c which makes F(x,y,z) a Conservative field 6) Using 9,6 and I found in part (a), find the potential function of F(x₁y, z). =) calculate & f.dr on the curve of intersection of C: 2=2x-3y and 2 = x² + 4y² from (0,0,0) to (1,1,2).

Q: 3. (a) Determine the value of coefficient a for which the vector field F = (ayz + 22, 2.02?, 6ryz² +…

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Q: 20) For each vector field F, evaluate the circulation along the curve C consisting of the portion of…

A: Step 1:Step 2:Step 3:Step 4:

Q: Find the directional derative of f at the point P0(1,1) in the direction of the vector A = 3i - 2j…

A: Direction derivatives at (1,1) is given in direction of A = 3i - 2j.

Q: Consider vector field F(x, y) = (axy - y³)i + (4y + 3x² + bxy²)j. (i) Determine the value of…

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Q: Consider a conservative vector field F(x, y) = (y + 6xy^3 )i + (x + y + 9x^2 y^2 )j (a) Find a…

A: Given: A conservation vector field F(x,y)=y+6xy3i^+x+y+9x2y2j^. To findf=∇f.

Q: Show that (vector)F = is a gradient field. Then find a potential function f for (vector) F.

A: A vector field F(x,y)=f1,f2 is a gradient field if ∂f2∂x=∂f1∂y

Q: (Section 17.3) Given the vector field F(x, y, z) = (lnx + sec²(x+y)) î+ (sec²( (sec² (r- (a) Show…

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Q: (b) Verify your answer in (a) using Stokes' theorem, where the normal vector to the surface has a…

A: We have to apply Stoke's theorem.

Q: b) Show the force field (x, y, z) = (x²y + In(yz)) i+ + y cos(2z) ) j+ y? sin(2z) 3 F k is a…

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Q: (a) Show that each of the vector fields F = 4yi + 4x j, Ğ = 24² +247 3₁ and H = √2+1+34 are gradient…

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Q: (a) Is the vector field F(x, y) = (xe³, ye²) conservative? Justify your answer. If it is…

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Q: af of H.W: Find and for the functions: b əx ду 1. f(x, y) = (xy-1)² 3. f(x, y) = (x²-1)(y + 2)…

A: Given functions are 1. f(x, y)=(xy-1)2 3. f(x, y)=(x2-1)(y+2) 4. f(x, y)=x(x2+y2)   To Find: ∂f∂x…

Q: Problem #6: Consider the vector field F(x, y) = (Axy+ 4xy + By)i+ (x* + 2x2 + 8xy)j (a) Find the…

A: A vector field is said to be conservative if it is irrotational. Also, if a vector field F→ is…

Q: Suppose F(x,y)=. Use Green's Theorem to calculate the circulation of F around the perimeter of the…

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Q: Compute the work done performed by the force F(x, y) = (4r'y", zyt) acting on the object moving…

A: workdone   by a force F  along curve C is∫C F dr Here F=4x2y3,xy4since c is closed curve in xy…

Q: Consider the motion p(a, t) given by p(x, t) = ((1+t)α₁, a2 + ta3, -ta₂+ α3)² and let f(x, t) = tx₁…

A: Given the motion φα,t=1+tα1, α2+tα3, -tα2+α3T and fx,t=tx1+x2 be an Eulerian scalar field.…

Q: Calculate Vf, V•f, V×f f (x,y. =)= xi+ y'j-* by the function

A: Introduction: The curl of the vector f(x,y,z)=fxi+fyj+fzk is given by ∇×f=∂∂xi+∂∂yj+∂∂zk×fxi+fyj+fzk…

Q: Let f(x, y) = ln (1 + x² + y²), then a) f is not differentiable at (0, 0). b) Exists a point (xo,…

A: We are given the function fx, y=ln1+x2+y4. Our aim is to choose the correct option for the given…

Q: Q1) Determine the gradient of the following scalar field V = e(2x+3y)cos5z Q2) Given f = x²y + 3z +…

A: Since you have asked multiple question, we will solve the first question for you.  If you want any…

Q: Find the gradient vector field of f. Sxy f(x,y) =xe

A: Here the given function is f(x,y)=xe5xy. We have to find the gradient vector field of the given…

Q: af Calculate at point (-1,2) where f (x, y) = 8+ x² + 3y? dv and v = i+j. 10 2 5/2 V2

A: fx, y=8+x2+3y2  &  v=i+j Partially differentiating with respect to x, we get fxx, y=0+2x+0=2x…

Q: For the scalar function f, div(grad f) is equal to Select one: a. + + dx?' ay? ' az? b. dx dy dz C.…

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Q: Consider the conservative force field F = (18x²y+9-y)i + x(6x² - 1)j. Find a function f(x,y) such…

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