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For the following fields, calculate div F.
1. F (x,y,z) = 〈2xz, -3xy, -z2〉
2. F (x,y,z) = 〈3x+8, -5y, 2z +1〉
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- If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xyLet where is a field and let . Prove that if is irreducible over , then is irreducible over .For the following fields, calculate curl F: 1. F(x,y,z) = 〈x2 -y, 4z, x2〉 2. F(x,y,z) - 〈-3y,3x〉
- Prove that polynomial x^2 + y^2 + y^2 − xy − yz − zx is irreducible in R [x, y].Let x, y ∈ F, where F is an ordered field. Suppose 0 < x < y. Show that x2 < y2.Evaluate the work done between point 1 and point 2 for the conservative field F.F = (y + z) i + x j + x k; P 1(0, 0, 0), P 2(9, 10, 8) a) W = 162 b) W = 0 c) W = 18 d) W = 90
- For the vector field F(x,y,z) = 2yi +xyj + (2x-xz)k. (i) Show that (ii) Determine whether if the vector field F is conservertive or not.find the divergence of the field. 1. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)k 2. F = (x ln y)i + (y ln z)j + (z ln x)kEvaluate the work done between point 1 and point 2 for the conservative field F.F = 6x i + 6y j + 6z k; P 1(4, 3, 5) , P 2(5, 6, 7) a)W = 180 b) W = 480 c) W = 0 d) W = - 180
- Use the zero curl test to determine whether the following fields are conservative: F = (2x/(x2 + y2))i + (2y/(x2 + y2))j + 2zkfind the divergence of the field. F = (x - y + z)i + (2x + y - z)j + (3x + 2y - 2z)kFind the divergence of the field F.F = 8x 5 i + 3y 5 j + 2z 5 k a) 65 b) 8x4 + 3y4 + 2z4 c) 0 d) 40x4 + 15y4 + 10z4