I cannot remember if I have already posted this question here... However,if that is the case,let it be said that I want to discuss more about it. Let us define 4 vector by 4 co-ordinates (x1,x2,x3,x4) where (x1,x2,x3) are space components (like x,y,z) and x4 is related to time as x4=ict.Express the following equations in tensor notation. (i)The continuity equation: div J+(del*rho/del t)=0 (ii)The wave equation del2ø-(1/c2) [del2 ø/del t2]=0 (iii)What will be the value of j4 instead of above? I have changed phi to psi... Do not mind. How can I get the 3rd part? For the first two part I am getting these: Please check... Let me know if I am correct. (i) (∂/∂x_i)(J_i)+ (∂/∂t)ρ=0 (ii) [∂^2/(∂x_i*∂x_i)]ψ-(1/c^2)[(∂^2/∂t^2)ψ]=0
I'll assume in (i) your reference of general delta is within the heat spectrum, and you're speaking of a gradient. Unless you meant change of work related forces within a system. If that's the case, ignore: Time span*Joules of E/ (-)Angular Momentum=Delta(sp.) In (ii)[assuming theta in the stead of NULL]: delta time=c^2=>time constant of light (any number of choices here). If assumptions withstanding, and by j4 your implying the unit time vector within a rotational sphere, I'd say your value would be: c^2=delta(sp.), which could lead to by assumption a plane reference, where time would drop out and you'd be left with a finite value of: xx*10^8. Or a span reference. Substitute units, as you may. Don't necessarily jump into infinite dwellings with partial integration, as these are typically reserved for very finite gaps within infinity. Algebraic solutions of sets to zero often suffice.
