Show whether or not F is a conservative field. If it is, find a potential function for it and use it to evaluate the line integral F dr along any path from (0,0) 6. to (1,1). F(x, y) = (2.xy + y)ĩ +(x² + x)}

Q: 5. Consider the force field F = (xy)i + (x −y)j and C, the straight line from the point (-1,2) to…

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

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Q: Is the vector field F(x, y) = (In y + sin(e²“), #

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Q: Consider the vector field 1 G(x, y, y, z) Cos y sin y 2√x' xsin y' 1+z² Show that G is conservative…

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Q: Consider F(x, y) = (y² − 2xy) i + (2xy — x²) j. (a) Show that F is a conservative vector field in…

A: the given vector field is F(x,y)=(y2-2xy)i+(2xy-x2)j we have to show that the vector field is…

Q: Evaluate the work done between point 1 and point 2 for the conservative field F. F 3D6xi+ 6уј+ 6z k;…

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Q: Find a potential field for F(x, y, z) = (y +z)i + (x + z²)j + (x+2yz)k or prove none exists.

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Q: 6. Consider the vector field defined by y F = (²) ² + ( 3y² m (x) - ₂ √ √ ) ³ + ( √T)k i j (a) Use…

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Q: Show that the vector field sin 2y x² Ĥ(x, y) = (y) = (₂ Y x² + 1 is conservative by finding all its…

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Q: 5. Is the following vector field F(r, y) = (2x – y, -x + 3y?) conservative? If so, find a potential…

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Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

A: F = ( Cosx )i + ( -y Siny )j + ( z3 )k  F = gradient ( - Sinx - y Siny - Cosy + z4/4 ). Thus, we can…

Q: Briefly describe how to find a potential function φ for a conservativevector field F = ⟨ƒ, g⟩ .

A: Before we get into the procedure to find the potential function, we need to understand certain…

Q: 11. LetF(x, y) = (ye™ – 6x))i + (xe™v + 4)7 (a) Show that F~ is a conservative vector field. (b)…

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Q: Find the gradient vector field (F(x, y, z)) of ƒ(x, y, z) = x³y³ z z F(x, y, z) =

A: Solution :-      The given function is     f(x,y,z) =  x6y3z4   then , we have to find gradient…

Q: 4. Let F(x, y) = (x², y²). Let C be the line segment from (0,0) to (1, 1), which lies on the vector…

A: As per the guideline I have solved first three subpart of the question.

Q: Let f(z) := z5 . Show that f is invertible in a neighbourhood of i and calculate (f-1)'(f(i)).

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Q: For each of the conservative vector fields below, find a potential function f. (1) F = 6yzi + 6xzj +…

A: a) To find a potential function f for the conservative vector field F = 6yzi + 6xzj + 6xyk, we need…

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: Find the potential function f for the field F. F = (y - z) i + (x + 2y - z) j - (x + y) k   a)…

A: Let us find the potential function

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

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Q: y +3z x + 2z 2y + 3x Consider the conservative vector field F(x, y, z) = with domain R³. Find a…

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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…

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Q: 6. Consider the vector field defined by F = ( 1² ) ² + ( 3,³² m (2). X 2 2√y (a) Use the component…

A: Since F is not conservative so we can not solve the part b and c.

Q: Find the gradient field of the function, f(x,y,z) = (2x² + 2y² +2²) -1/2

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Q: Calculate the work done by the field F (x, y, z) = (ex, e, xyz) when the object moves along the path…

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Q: Consider the force field F = (3x²yz - y + z, x³z - x, x³y + x). (a) Verify that it is a conservative…

A: A vector field will be conservative if its curl is zero . If f is potential function of F then ∫abF…

Q: 19. Find potential function for = (3xy, æ³)

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Q: Prove that F = (y' cosx+z')î +(2ysin x-4)ĵ+(3xz² +2)k is a conservative force field, find its scalar…

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