Shortest Line in 3D Prove that the shortest path between two points in 5. three dimensions is a straight line. Hint: Write the path in the parametric form x = x(u), y use the Euler-Lagrange equations. y(u), and z = 2(u). Then
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- A triangle in the xy plane is defined with corners at (x, y) = (0,0), (0, 2) and (4, 2). We want to integrate some function f(x, y) over the interior of this triangle. Choosing dx as the inner integral, the required expression to integrate is given by: Select one: o Sro S-o f(x, y) dx dy x=0 2y y=0 O S-o So F(x, y) dæ dy O o S f(x, y) dy dæ O So So F(x, y) dx dy x/2 =0Q. Derive the relation betueen unit Vectors of Cylindrical and Carterian Coordinates?Problem 3: Two-level system and density matrice Suppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written as X = a000 + a.σ, where ao and ak, k = 1, 2, 3, are numbers, 0o = 1 is the identity matrix and o are the Pauli matrices. (a) How are ao and a related to tr(X) and tr(OX)? Obtain ao and ak in terms of the matrix elements Xij. Assume that ao, ak ER such that X is Hermitian and could be interpreted as a Hamiltonian, what are the eigenvalues of X?
- (e) v,y &+(2xy+)9+2yz &. Problem 16 Sketch the vector function %3D and compute its divergence. The answer may surprise you...can you explain it?b) Given F(x, y, z) = (x³ + cosh z) i+ (2y³ – 3r²y)j – (x² + 4y²z) k. Use Gauss's theorem to calculate //F .n dS where n is the outward unit normal of o, the surface bounded by the planes, x = 0, z = 0 and x + z = 6, and the parabolic cylinder x = 4 – y².Find the gradient field F = Vo for the potential function o below. P(x,y.z) = In (3x +y° +z²) Vøxy.z) = (O Vo(x.y.z z)3D
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