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Assgn1 - Assignment#1 Vector Algebra 1 Three field...

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Assignment #1 – Vector Algebra 1) Three field quantities are given by P = 2 a x - a z Q = a x - a y + 2 a z R = 2 a x - 3 a y + a z Determine (a) An unit vector perpendicular to both Q and R (b) The component of P along Q (c) (P+Q) x (P-Q) (d) Q. R x P 2) Given the vector field (cylindrical coordinates) H = ρ z cos φ a ρ + sin ( φ /2) a φ + ρ 2 a z At point (1, π /3,0), find (a) H . a x (b) H x a θ (c) The vector component of H normal to surface ρ =1 (d) The scalar compenent of H tangential to the plane z=0 3) Calculate (a) the angle between the normals to the surfaces x 2 y+z = 3 and x logz –y 2 =-4 at the point of intersection (-1,2,1) (b) the angle at which the line x=y=2z intersects with the ellipsoid x 2 + y 2 + 2z 2 =10 4) Determine (a) the divergence of these vector fields P = x 2 yz a x +xz a z Q = ρ sin φ a ρ + ρ 2 z a φ + z cos φ a z (b) the flux of D = ρ 2 cos 2 φ a ρ + z sin φ a φ over the closed surface of the cylinder 0 z 1, ρ =4.Verify the divergence theorem for this case.
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