 #1
DottZakapa
 239
 17
 Homework Statement:

The outflux of the vector field
F(x,y,z) = (sin(2x) + ye3z,(y + 1)2,−2z(y + cos(2x) + 3)
from the domain
D = {(x, y, z) ∈ R^3 : x^2 + y^2 + z^2 ≤ 1, x ≥ 0, y ≤ 0, z ≥ 0}
 Relevant Equations:
 flux of a vector field
My idea is to evaluate it using gauss theorem/divergence theorem.
so the divergence would be
## divF = (\cos (2x)2+2y+22z ( y+\cos (2x)+3) ) ##
is it correct?
In this way i'ma able to compute a triple integral on the volume given by the domain
## D = \left\{ (x, y, z) ∈ R^3 : x^2 + y^2 + z^2 ≤ 1, x ≥ 0, y ≤ 0, z ≥ 0 \right\} ##
then using spherical coordinates
##x= r cos\theta \sin\phi\\
y= r\sin\theta\sin\phi\\
z=r\sin\phi##
with the following boundaries
##0\leq\phi\leq\frac \pi 2##
##\frac {3\pi} {2}\leq\theta\leq 0##
then i do the substitution in the divergence and solve the integral
##\iiint_V divF r^2\sin\phi dr d\theta d\phi ##
am i doing it correctly or is there anything wrong?
so the divergence would be
## divF = (\cos (2x)2+2y+22z ( y+\cos (2x)+3) ) ##
is it correct?
In this way i'ma able to compute a triple integral on the volume given by the domain
## D = \left\{ (x, y, z) ∈ R^3 : x^2 + y^2 + z^2 ≤ 1, x ≥ 0, y ≤ 0, z ≥ 0 \right\} ##
then using spherical coordinates
##x= r cos\theta \sin\phi\\
y= r\sin\theta\sin\phi\\
z=r\sin\phi##
with the following boundaries
##0\leq\phi\leq\frac \pi 2##
##\frac {3\pi} {2}\leq\theta\leq 0##
then i do the substitution in the divergence and solve the integral
##\iiint_V divF r^2\sin\phi dr d\theta d\phi ##
am i doing it correctly or is there anything wrong?