4. Calculate the commutator (ar2 fx] where a is a constant and r = 7x2 +...
3. (10 points) Let X be continuous random variable with probability density function: fx(x) = 7x2 for 1<<2 Compute the expectation and variance of X 4. (10 points) Let X be a discrete random variable uniformly distributed on the integers 1.... , n and Y on the integers 1,...,m. Where 0 < n S m are integers. Assume X and Y are independent. Compute the probability X-Y. Compute E[x-Y.
4. Rank-2 tensors A charge q moves with a constant velocity u ux. Define the antisymmetric tensor r" by r"-1 (ημ2-η"z"), where ημ įs the four velocity of the charge. Further define (a) Show that r-lauP + y2 + z2. -tu (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the s,y or z component of E where the Imc (r2) TC
4. Rank-2 tensors A charge q moves with a...
Use spherical coordinates to calculate the triple integral of fx, y, z) over the given region. rx, y, z) = ρ; x2 + y2 + Z2 16, 2.52, x20
Use spherical coordinates to calculate the triple integral of fx, y, z) over the given region. rx, y, z) = ρ; x2 + y2 + Z2 16, 2.52, x20
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
4) (5 marks) Using the method demonstrated in class, find the general potential, U(r), for F -kr where k is a constant, and express it in terms of r = VX2 + y2 + Z2
3. If z = f(x,y), where x = r cos, y=r sin 0 show that 222 222 1 222 1.az + + +) ar2 ду? ar2 a02 rar
4. (15 pts) Evaluate I = SSSR (x2 + y2 + z²) dV where R is the cylinder given by 4 < x2 + y2 +z? 59.
Calculate (r is the usual position vector: (cose/r?), 7(rr), Yx(FxB) where B is constant vector along z direction.
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the
4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the
4. Rank-2 tensors A charge q moves...
4. Rank-2 tensors A charge q moves with a constant velocity tensor rby rHv 2 u. Define the antisymmetric ) where r" is the four velocity of the charge. Further define u (n" =-rH"ru/2. (x-ut)2 1-u2/c2 + y2 + z2 |(a) Show that r2 (b) Show that the electric field produced by the charge is given by En/c subscript n refers to the x,y or z component of E 4TC (r2)3/2 Where the
4. Rank-2 tensors A charge q moves...