find `(del z)/(del t)` and `(del z)/(del s)` if `z= x^3+y^3+2x^2y` where `x=2t+s` and `y=t-2s`

If u=x^y then find (del u)/(del x) and (del u)/(del y) .

If z (x + y) = x ^ (2) + y ^ (2) show that [(del z) / (del x) - (del z) / (del y)] ^ (2) = 4 [1 - (del z) / (del x) - (del z) / (del y)]

Solve the partial differential equation (del z)/(del y)=xy :

Show that the equation, y=a sin(omegat-kx) satisfies the wave equation (del^(2) y)/(delt^(2))=v^(2) (del^(2)y)/(delx^(2)) . Find speed of wave and the direction in which it is travelling.

Making use of the Carnot theorem, show that in the same of physically uniform substance whose state is defined by the parameters T and V (del U//delV)_T = T(del p//del T)_v - p where U(T, V) is the internal energy of the substance. Consider the infinitesmal Carnot cycle in the variables p, V .

(i) In a travelling sinusoidal longitudinal wave, the displacement of particle of medium is represented by s = S (x, t) . The midpoint of a compression zone and an adjacent rarefaction zone are represented by letter ‘C’ and ‘R’ respectively. The difference in pressure at ‘C’ and ‘R’ is DeltaP and the bulk modulus of the medium is B. (a) How is |(del s)/(del x)| related to |del/del| (b) Write the value of |(del s)/(del x)|_(C) in terms of DeltaP and B . (c) What is speed of a medium particle located mid-way between ‘C’ and ‘R’. (ii) A standing wave in a pipe with a length of L = 3 m is described by s=A cos ((3pi x)/(L)) sin ((3pi vt)/(L)) where v is wave speed. The atmospheric pressure and density are P_(0) and rho respectively. (a) At t=L/(18v) the acoustic pressure at x =L/2 is 0.2 percent of the atmospheric pressure . Find the displacement amplitude A. (b) In which overtone is the pipe oscillating?

If x = r cos theta and y = r sin theta, prove that (del ^ (2) r) / (del x ^ (2)) + (del ^ (2) r) / (del y ^ (2)) = (1) / (r) [((del r) / (del x)) ^ (2) + ((del r) / (del y)) ^ (2)]

If U=(y)/(z)+(z)/(x), then find x(del u)/(del x)+y(del u)/(del y)+z(del u)/(del z) .

If u = x ^ (2) tan ^ (- 1) ((y) / (x)) - y ^ (2) tan ^ (- 1) ((x) / (y)), prove that (del ^ (2) u) / (del x del y) = (x ^ (2) -y ^ (2)) / (x ^ (2) + y ^ (2))

The x and y coordinates of a particle at any time t are given by x = 2t + 4t^2 and y = 5t , where x and y are in metre and t in second. The acceleration of the particle at t = 5 s is