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A: .
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- Δ denotes the Laplace operator defined by Δφ = ∂2φ/∂x2 + ∂2φ/∂y2 + ∂2φ/∂z2 . Show that if F is the gradient of a harmonic function, then curl(F) = 0 and div(F) = 0.find the curl of U = exy ax +sin(xy)ay + cos2(xz) azSketch the direction field of the differential equa tion.Then use it to sketch a solution curve that passes through thegiven point. y' = x - xy , (1,0)
- Find the gradient vector field for the scalar function f(x,y)=sin(2x)cos(6y). Enter the exact answer in component form. ∇(x,y)=Sketch the direction field of the differential equa tion.Then use it to sketch a solution curve that passes through thegiven point. y' = y - 2x , (1,0)Suppose C is the curve from (0, 0) to (2, 0) to (2, 3) to (0, 3) to (0, 0). Find the work done by the vector fieldF(x, y) = <x^(3)−2y^(2), x + cos(√y)> on a particle moving along C.
- Evaluate a 3d divergence of 1/r^2 in the radial directionIntegrate ƒ(x, y) = sqrt(4 - x2) over the smaller sector cut from the disk x2 + y2 <=4 by the rays u = pai/6 and u = pai/2.Find the work done by the force field F(x,y) = 4yi + 2xj in moving a particle along acirclex^2 + y^2 =1 from (0, 1) to (1, 0).
- or this problem, consider a particle traveling within the force field F = < -y,x,1/2 > along the parametrized curve r(t) = < t cos(t),t sin(t),1/2t > from the point (0,0,0) to the point (2pi,0,pi) Explain why the work done moving the particle along the path in this force field is positive. Compute the work done on a particle traveling along the given parametrized curve within the force field.determine laplace of L{5 - 3t + 4*sin(2t) - 6*exp(4t)}Find a linear approximation in order to approximate the change in f(x, y, z) = e4x cos(5yz) when moving from the origin a distance of 0.3 units in a direction of <2, 3, 1>