For R the vector between the source coordinate position Pxyz

For R the vector between the source coordinate position P(x’,y’,z’) and the field point O(x,y,z) whose magnitude is R= ((x-x’)^2+(y-y’)^2+(z-z’)^2)^½ evaluate

‘(1/R) where ’=(d/dx’,d/dy’,d/dz’) is the gradient with respect to the source coordinates.

Solution

R = O - P

R = (x - x\')i + (y-y\')j + (z-z\')k

Or R = ( x-x\' , y-y\' , z-z\' )

delta\' (1/R) = (d(1/x-x\') / dx\' , d(1/y-y\') / dx\' , d(1/z-z\') / dx\' )

= (-ln(x-x\') , -ln(y-y\') , -ln(z-z\') )