The work differential, `dw`, is

If w(x,y,z) = x^2y+y^2z + z^2x,x,y,z in R, find the differential dw.

If W = x ^(2) + y ^(2) + z^(2), x,y,z in R find the differential dW.

If u, v, w are three differentiable functions of x , prove that d/dx (uvw) = ((du)/dx)vw + u ((dv)/dx) w + uv ((dw)/dx) . Use this result to differentiate x^(3)(x^(2)+1)(x^(4)+1) w.r.t. x.

A sample of an ideal gas undergoes on isothermal expansion. If dQ, dU and dW represent the amount of heat supplied, the change in internal energy and the work done respectively,then

If u, v and w are functions of x, then show that d/dx(u cdot v cdot w) = du/dx (v cdotw + u cdot dv/dx cdot w +u cdot v dw/dx) in two ways - first by repeated application of product rule, second by logarithmic differentiation.

If u,v and w are functions of x ,then show that (d)/(dx)(u*v*w)=(du)/(dx)v*w+u*(dv)/(dx)*w+u*v(dw)/(dx) in two ways - first by repeated application of product rule,second by logarithmic differentiation.