Show that the vector field F = (yz² ye, rz² - e, 2xyz) is conservative, and find a potential function for it.
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- Determine whether the vector field h(x,y) = y^2i + (2xy-e^2y)j is conservative or not. If it is conservative, determine a potential functionFor the vector field F = (1/z)I -4j -(x/z^2)k , find the potential function f for the field.Choose a correct potential function for the conservative vector fieldF = (3x-y+1, –x-4y-2).
- Calculate the line integral of the vector field F = -3 i+ 2j along the line from the point (4, 0) to the point (12, 0). The line integral =Find the directional derivative of the scalar field þ= x¹y + 4xz at the point (2, -2, 4) along the direction vector (2, -1, -2):Express the vector field D = ( x2 + y2 )-1 ( xax + yay ) in cylindrical components and cylindrical variables.
- determine if the given vector field f is conservative and hence find the potential function for the vector field if it is conservative F=(6x^2-2xy^2+y/2√x) i-(2x^2y-4-√x) j.Consider the vector field v(x)=(y/(x^2+y^2))i − (x/(x^2+y^2)) j + 4k. Calculate ∫Cv dr, where C is the straight line from (1,0,1)to (0,1,2).Find the gradient vector field of f(r, y) = In(x + 3y)