Tensors: Defining & Expressing Equations

[neelakash]

Homework Statement

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 del2ψ-(1/c2) [del2 ψ/del t2]=0

(iii)What will be the value of j4 instead of above?

Homework Equations

The Attempt at a Solution

I got totally stuck.Please help me to start with it.

 

[nrqed]

Homework Statement

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 del2ψ-(1/c2) [del2 ψ/del t2]=0

(iii)What will be the value of j4 instead of above?

Homework Equations

The Attempt at a Solution

I got totally stuck.Please help me to start with it.

Just rewrite all th epartial derivatives with respect to x,y,z and t in terms of partial derivatives with respect to x1, x2, x3, x4.

Of course, for the first three, it's trivial, i.e.

$\frac{\partial}{\partial x} = \frac{\partial}{\partial x_1}$ etc. For the time part, you will have a factor of ic appearing in the relation. Now, just replace all your partial derivatives in your equations.