What is Volume integrals: Definition and 14 Discussions

In Vanderlinde page 171-172, the author derives the vector potential for the magnetic dipole (and free currents) \begin{align} \vec{A}(\vec{r}) &=\frac{\mu_{0}}{4 \pi} \int_{\tau} \frac{\vec{J}\left(\vec{r}^{\prime}\right) d^{3}...

I tried integrating the 4-volume of a 4-hemisphere, that is, $$\int^{R}_{0} \frac{4}{3} \pi r^3 dw$$ (along w-axis), since ##r## is proportional to ##w##, where ##r=\frac{w}{R} R##, ##r=w##, thus the integral becomes $$\int^{R}_{0} \frac{4}{3} \pi w^3 dw = \frac{\pi}{3} R^4$$ The volume of a 4-D...

Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##. Consider the following multiple integral: ##\displaystyle A=\iiint_{V'} \left[ \iint_S \dfrac{\cos(\hat{R},\hat{n})}{R^2} dS \right] \rho'\ dV' =4 \pi\ m_s## where...

Consider a one to one transformation of a ##3##-##D## volume from variable ##(x,y,z)## to ##(t,u,v)##: ##\iiint_V dx\ dy\ dz=\int_{v_1}^{v_2}\int_{u_1}^{u_2}\int_{t_1}^{t_2} \dfrac{\partial(x,y,z)}{\partial(t,u,v)} dt\ du\ dv## ##(1)## Now for a particular three dimensional volume, is it...

Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...

If I want to integrate the volume inside a cylinder ##x^2+y^2 = 4R^2##, and between the plane (I think it's a plane) ##z= \frac{x^2+3y^2}{R}## and the xy plane, then I know how to convert it to cylindrical co-ords, find the limits of integration, and integrate r dr dθ dz. But exactly what am I...

Homework Statement (3 i) Using \nabla . \mathbf{F} = \frac{\partial \mathbf{F_{\rho}}}{\partial \rho} + \frac{\mathbf{F_{\rho}}}{\rho} + \frac{1}{\rho} \frac{\partial \mathbf{F_{\phi}}}{\partial \phi} + \frac{\partial \mathbf{F_{z}}}{\partial z} calculate the divergence of the vector field...

Homework Statement Evaluate the integral as either a volume integral of a surface integral, whichever is easier. \iiint \nabla .F\,d\tau over the region x^2+y^2+z^2 \leq 25, where F=(x^2+y^2+z^2)(x*i+y*j+z*k) Homework Equations \iiint \nabla .F\,d\tau =\iint F.n\,d\sigma The...

So I am trying to understand how and why the limits of surface and volume integrals come about. I think I came up with a easy to understand argument but not a mathematically sound one. Frankly its a little dodgy. Can anyone provide feedback on this argument or provide a better and possibly more...

Help choose the limits of the following volume integrals: 1) V is the region bounded by the planes x=0,y=0,z=2 and the surface z=x^2 + y^2 lying the positive quadrant. I need the limits in terms of x first, then y then z AND z first, then y and then x. And also polar coordinates, x=rcost...

In another forum, I have been challenged to prove mathematically that a certain idea which consists of fields of discrete elements will satisfy http://en.wikipedia.org/wiki/Divergence_theorem" . The fields are not expressible in terms of a differentiable function but rather consist of discrete...

Please help me check if the following reasoning is correct: When considering line and surface integrals, one must integrate over a scalar or vector field. The infinitesimal line (dl) or surface (dA) segments can be treated either as vectors or scalars. Therefore, the only types of line and...

Hey guys, could one of you explain why when doing a volume integral using spherical polar coordinates, you have the limits as 2 pi to 0 on phi but only pi to 0 on theta? Thanks. To clarify, I've been doing this all this time for questions, but it just occurred to me that I Don't know why i do...