If `u=x^y`, show that `(del^2u)/(dely delx)`= `(del^2u)/(delxdely)`

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

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

If u= tan^-1((y^5)/(x^2)) , then find i. x((delu)/(delx)) + y((delu)/(dely)) ii. x^2((del^2u)/(delx^2))+2xy((del^2u)/(delxdely))+y^2((del^2u)/(dely^2))

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.

If z(x+y)= x^2 +y^2 show that (frac{delz}{delx}-frac{del z}{del y})^2 = 4 (1- frac{delz}{delx}+frac{delz}{dely})

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

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v,v),(u'',v'',2v'):}| .

The function u=e^x sin x ; v=e^x cos x satisfy the equation a. v(d u)/(dx)-u(d v)/(dx)=u^2+v^2 b. (d^2u)/(dx^2)=2v c. (d^2v)/(dx^2)=-2u d. (d u)/(dx)+(d v)/(dx)=2v

If (x+i y)^3=u+i v , then show that u/x+v/y=4(x^2-y^2) .

If (x+i y)^3=u+i v , then show that u/x+v/y=4(x^2-y^2) .