I am looking to derive general formulas for the electric fields generated by general charged objects given their charge densities. For linear and surface charge densities I have been able to derive the following expressions for a linear charge density $\lambda(\vec{r})$ and surface charge density $\eta(\vec{r})$, respectively:
$$\vec{E}=\frac{1}{4\pi\varepsilon_0}\int_C\frac{\lambda(\vec{r})}{\Vert\mathbf x_2-\mathbf x_1\Vert^3}(\mathbf x_2-\mathbf x_1)\,ds$$
$$\vec{E}=\frac{1}{4\pi\varepsilon_0}\iint_S\frac{\eta(\vec{r})}{\Vert\mathbf x_2-\mathbf x_1\Vert^3}(\mathbf x_2-\mathbf x_1)\,dS$$
I am able to compute these given that I am able to find a parameterization of the curve $C$ or the surface $S$, using the following formulas:
$$\int_C f\,ds=\int_a^b f\big(\vec{r}(t)\big)\Vert\vec{r}'(t)\Vert\,dt$$
$$\iint_S f\,dS=\int_c^d\int_a^b f\big(\vec{r}(u,v)\big)\Vert\partial_u\vec{r}\times\partial_v\vec{r}\Vert\,du\,dv$$
However, I am uncertain how to do this in the case with a volume charge density $\rho(\vec r)$. I do not know a formula to simplify the volume integral. My guess would be maybe something like:
$$\iiint_Rf\,dV=\int_\alpha^\beta\int_\gamma^\delta\int_\epsilon^\zeta f\big(\vec r(u,v,w)\big)\Vert\partial_u\vec r\times\partial_v\vec{r}\times\partial_w\vec{r}\Vert\,du\,dv\,dw$$
But I do not know a method to derive this. Is there a method/formula for computing these types of integrals given a parameterization of the region $V\subset\mathbf{R}^3$?
Best Answer
For a volume integral over an open set $U \subseteq \mathbb{R}^3$, the expression is pretty straightforward. Since this question is physics related, I will introduce coordinates into the expression in a very explicit way. Let $\phi: U \to \mathbb{R}^3$ denote the coordinate map. The volume integral is then given by
$$\int\int\int_UfdV := \int \int \int_{\phi(U)}(f \circ \phi^{-1})Jdq_1dq_2dq_3$$
where $J$ denotes the Jacobian determinant. Integration on the right-hand-side is a triple integral on the image of $U$ under $\phi$.