Consider vector field F(x, y) = (axy - y³)i + (4y + 3x² + bxy²)j. (i) Determine the value of constant a and b so F would be conservative vector field. (ii) Determine its potential function. (iii) Then, find the work done by F to move one particle from (1,0) to point (0,1).

Q: Find the directional derivative of the scalar field = x¹y + 4xz at the point (2, -2, 4) along the…

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Q: F(x,y) = x²yi + 3y³x²j Find the integral curves for the vector field Оа. -бу + 1 = C 6ху? b. -6ху —…

A: given vector field Fx,y=x2y i^+3y3x2 j^ claim- find the integral curves for the vector field

Q: Find the directional derivative of the scalar field = x³y + 4xz at the point (1,-1, 2) along the…

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Q: Let r(t) = 5e7t + e2t and y(t) = 3e2t + 1le (i) Rewrite the given functions in the vector form Y(t)…

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Q: The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end…

A: Given, vector field F→x,y=2xy,x2 and three curves that start at 1,2 and end at 3,2.

Q: F = (x² - y)i + + (4z)j + (x²)k Find the curl (curl calculation) of the vector field?

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Q: Find the gradient vector field of f(r,y, z) = x*y* +y³z³ + æz°.

A: The gradient vector field of f(x,y,z) is given by ∇f=∂∂xf(x,y,z) i^+∂∂yf(x,y,z) j^+∂∂zf(x,y,z) k^…

Q: Find a potential function of the vector field. F(x,y) = (3x²y + 1, x³+2y)

A: we have to find potential function.

Q: Let R be a differentiable vector-valued function such that R'(t) = (3t2, 2tvt2 + 1, 2t + 3) and R(1)…

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Q: The gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² »i O II. (4xy)i…

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Q: ed by F(x, y) = (-: %3D

A: The function given is , F→x,y=-x,y. Draw the table for the points plot on the graph for the vector…

Q: A particle traveling in a straight line is located at the point (1, -1, 2) and has speed 2 at time t…

A: Given: A particle traveling in a straight line is located at the point (1, -1, 2) and has speed 2 at…

Q: f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?

A: To check whether it is a curve, vector or surface. f(x,y) = 2x3+4y3-3x2-120x-192y+5

Q: 1) Sketch the vector field F. (i) F(r, y) = (ii) F(r,y, 2) = i

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Q: Find the derivative of the vector function r(t) = ta x (b+ tc), where a = (4, -1,5), b = (-3,-3,-1),…

A: Given, the vector function      r(t)=ta×(b+tc),where       a=4,-1, 5, b=-3,-3, -1, and  c=1, 2, 2,…

Q: Find a potential function for the vector field .F(x,y)=

A: The vector field is

Q: Calculate the work done by the force field on the particle.

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Q: A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector…

A: To Determine: A net is dipped in a river .determine the flow rate of water across the net if the…

Q: Plot the vector field F = (y,sinx) using the viewing window (-2pi,2pi)x(-2pi,2pi

A: Given:       F = ( y , sin(x) )      viewing window :  (-2pi  ,  2pi)

Q: Find the upward flux of F= (14e2,-7e²,-6) through the first-octant part of the triangle a+2y+3z-6.…

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Q: Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical…

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