Find whether the given field vector is: i) conservative, ii) irrotational, iii) both? Justify your answer. Given, the vector is, V=yzî+ xzj+ xyk.
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- Calculate the flux of vector field F = (xy°, x²y) across the circle of radius 1 centered at coordinates (0, –1).Given a charge q1=50nC located at the XY coordinates (5m, 10m) and a charge q2= -20nC located at the XY coordinates (-3m, -1m). Find the field point has coordinates (-3m. 4m). Find the unit vector r1_ hat and r2_hat?consider the parallelepiped with sides: A=3i+2j+k، B=i+j+2k, c=i+3j+3k, then 1-Find the rolume of the paralldepiped 2-Find the area of the face determined by A and B. 3-Find the angle between the vactor C and the plane containing the face determined by A and B
- Calculate the flux of the given vector field by evaluating the line integral directly alongthe given curve for the below parts:(a) The vector field is ⃗ F = (x − y)⃗i + x⃗j. The curve is the circle x^2 + y^2 = 1in the xy-plane. Use the parameterization x = cos t and y = sin t.(b) The vector field is ⃗ F = (x − 1)⃗i + y⃗j. The curve is a circle of radius 3centered at (1, 1). The parametric form of this circle is⃗r = (1 + 3 cos t)⃗i + (1 + 3 sin t)⃗j, 0 ≤ t ≤ 2π(c) The vector field is ⃗F = x⃗i + y⃗j. The curve is the line segment from thepoint (0, 1) to the point (1, 3).A vector field is pointed along the z-axis, v → = α /x2 + y2 z ^ . (a) Find the flux of the vector field through a rectangle in the xy-plane between a < x < b and c < y < d . (b) Do the same through a rectangle in the yz-plane between a < z < b and c < y < d . (Leave your answer as an integral.)Evaluate the surface integral F⚫ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = −xi - yj + z³k, S is the part of the cone z = √x² + y² between the planes z = 1 and z = 4 with downward orientation ZA z = √√x² + y² xx 0 z=4