 #1
 112
 18
Homework Statement:

The vector field ##\vec F= 2x^2y \hat i  y^2 \hat j + 4xz^2 \hat k ## is defined over the region in the first octant bounded by ## y^2+z^2=9## and x=2. Find the value of ##\iint_S \vec {F} \cdot \hat n dS##
(a)100
(b)18
(c)0.18
(d)1.8
Relevant Equations:
 Gauss's Divergence Theorem.
##\vec F= 2x^2y \hat i  y^2 \hat j + 4xz^2 \hat k ##
## \Rightarrow \vec \nabla \cdot \vec F= 4xy2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume integral of the divergence.
Flux = ##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} \int_{\rho=0}^{3} (4h \rho cos \phi 2 \rho cos \phi + 8h \rho sin \phi ) \rho d \rho d \phi dh##
=##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} \int_{\rho=0}^{3} (4h cos \phi 2 cos \phi + 8h sin \phi ) \rho^2 d \rho d \phi dh##
=##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} (4h cos \phi 2 cos \phi + 8h sin \phi ) [\rho^3 /3]_0^{3} d \phi dh##
=##9\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} (4h cos \phi 2 cos \phi + 8h sin \phi ) d \phi dh##
=##9\int_{\phi=0}^{\pi /2} [2h^2 cos \phi 2h cos \phi + 4h^2 sin \phi ]_0^2 d \phi##
=##9\int_{\phi=0}^{\pi /2} (8cos \phi 4cos \phi + 16 sin \phi)d \phi##
=##36\int_{\phi=0}^{\pi /2} (cos \phi +4 sin \phi)d \phi##
=##36( [sin \phi ]_0^{\pi/2} +4 [cos \phi ]_{\pi/2}^0 )##
=36(1+4) = 36*5 = 180
Which doesn't match anything. The answer given is (a) 100.
Is there something which I did wrong here?
## \Rightarrow \vec \nabla \cdot \vec F= 4xy2y+8xz##
Let's shift to a rotated cylindrical system with axis on x axis:
##x \to h, y \to \rho cos \phi, z \to \rho sin \phi ##
Then our flux, as given by the Divergence theorem is the volume integral of the divergence.
Flux = ##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} \int_{\rho=0}^{3} (4h \rho cos \phi 2 \rho cos \phi + 8h \rho sin \phi ) \rho d \rho d \phi dh##
=##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} \int_{\rho=0}^{3} (4h cos \phi 2 cos \phi + 8h sin \phi ) \rho^2 d \rho d \phi dh##
=##\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} (4h cos \phi 2 cos \phi + 8h sin \phi ) [\rho^3 /3]_0^{3} d \phi dh##
=##9\int_{\phi=0}^{\pi /2} \int_{h=0}^{2} (4h cos \phi 2 cos \phi + 8h sin \phi ) d \phi dh##
=##9\int_{\phi=0}^{\pi /2} [2h^2 cos \phi 2h cos \phi + 4h^2 sin \phi ]_0^2 d \phi##
=##9\int_{\phi=0}^{\pi /2} (8cos \phi 4cos \phi + 16 sin \phi)d \phi##
=##36\int_{\phi=0}^{\pi /2} (cos \phi +4 sin \phi)d \phi##
=##36( [sin \phi ]_0^{\pi/2} +4 [cos \phi ]_{\pi/2}^0 )##
=36(1+4) = 36*5 = 180
Which doesn't match anything. The answer given is (a) 100.
Is there something which I did wrong here?