A beautiful flux integral Consider the potential function

QUESTION:

beautiful flux integral Consider the potential function \(\varphi(x, y, z,)=G(\rho)\), where G is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}}\) therefore. G depends only on the distance from the origin.

a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\).

b. Let S be the sphere of radius a centered at the origin and let D be.the region enclosed by S. Show that the flux of F across S is \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d A=4 \pi a^{2} G^{\prime}(a)\)

c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\)

d. Use pan (c) to show that the flux across S (as given in pan (b) ) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

Text Transcription:

varphi(x, y, z,) = G(p)

rho = sqrt x^2 + y^2 + z^2

F = nabla_varphi = G’(rho)r/rho

r = langle x, y, z rangle

rho = |r|

Nabla cdot F = nabla cdot nabla_varphi = 2G’(rho)/rho + G”(rho)

iint_D nabla cdot F dV