- #1

Kaguro

- 221

- 57

- 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= 4xy-2y+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= 4xy-2y+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?