Apashanka
- 427
- 15
- Homework Statement
- Three dimensional ##\delta## function
- Relevant Equations
- Three dimensional Delta function
##r,\theta,\phi## are the usual spherical polar coordinate system.
##\int_v\nabla•(\frac{\hat r}{r})dv## over a spherical volume of radius ##R## reduces to ##\int_s(\frac{\hat r}{r})•\vec ds=4\pi R##
Now ##r## runs from 0 to ##R,\theta## from 0 to ##\pi## and ##\phi## from 0 to ##2\pi##.
In terms of ##\delta## function ##\int_{0}^{R} \delta (r-r')dr=1## where ##r'## lies within 0 to R ,and ##\ne 0,R##,similarly for ##\theta## and ##\phi## also where ##\int_{0}^{\pi} \delta(\theta-\theta')d\theta=1## where ##\theta'## lies ##{0,\pi}## and ##\ne0,\pi## similarly for ##\phi## also.
Putting them
##\int_v(\nabla•\frac{\hat r}{r})dv=4\pi R\int_0^R\int_0^\pi \int_0^{2\pi} \delta(r-r')\delta(\theta-\theta')\delta(\phi-\phi')drd\theta d\phi##
can anyone please tell me how can the RHS be reduced to volume integral form??
##\int_v\nabla•(\frac{\hat r}{r})dv## over a spherical volume of radius ##R## reduces to ##\int_s(\frac{\hat r}{r})•\vec ds=4\pi R##
Now ##r## runs from 0 to ##R,\theta## from 0 to ##\pi## and ##\phi## from 0 to ##2\pi##.
In terms of ##\delta## function ##\int_{0}^{R} \delta (r-r')dr=1## where ##r'## lies within 0 to R ,and ##\ne 0,R##,similarly for ##\theta## and ##\phi## also where ##\int_{0}^{\pi} \delta(\theta-\theta')d\theta=1## where ##\theta'## lies ##{0,\pi}## and ##\ne0,\pi## similarly for ##\phi## also.
Putting them
##\int_v(\nabla•\frac{\hat r}{r})dv=4\pi R\int_0^R\int_0^\pi \int_0^{2\pi} \delta(r-r')\delta(\theta-\theta')\delta(\phi-\phi')drd\theta d\phi##
can anyone please tell me how can the RHS be reduced to volume integral form??