corresponding value of x and log, (305,2.4843) Certain mre (300,2.4771), and (304,2.4829), (307,2.4871). Find log₁0 301 using interpolation by it cration.
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- Given vectors a=(5,3) and b=(-1,-2) Find the x-component of the resultant vector: T` = 2 a + 36Calculate the divergences of cach of the following vectors: (a) v = 3k (b) v = r (c) = (4xz + y?) î + (12a² – 2²) ĵ + (xy – yz) k (d) ở = -yî+ x} (c) v = . -Using spherical polar coordinates r, 0, p to find CM of a uniform solid hemisphere of radius R, whose flat face lies in the xy plane with its center at the origin. The element of volume is in spherical polars of dV = r² dr sine de dip.
- (3) The natural independent variables for U are (S, V), from dU = TdS – pdV. U = U(S,V) (), instead. Show that U = U (V,T) leads to a much more complicated expression for p, namely gives simple expressions for T and p as T = and ().: Suppose you use (V,T) p = - V dT + f(V), %D T where f (V) is an unknown function of V.Find the first and second derivatives of the function. y = e²xsin(5x) y' مسلم = y" =Let n 1 be an integer, let to to is given by a stationary path of the Lagrangian functional C: L[x] = 1 dt L(t,x,x), x(0) =x0, x(t)=X1, where LT - V and T is the total kinetic energy T = n 1 k=1 2 mark. Using the above first-integral, show that, if V is independent of t, the total energy E=T+V of the particle is a constant of the motion.