Question

calculate the surface integral of the vector r=(x,y,z) over the entire surface of the cylinder given by x^2+y^2=1, 0 14 years ago

Answer 1

Use Gauss's Divergence Theorm.

Answer 2

\[\int\limits \int\limits_{\S} (F.n) dS = \int\limits \int\limits_{V} \int\limits (divF) dV\]

Answer 3

F=(x,y,z)

divF = (1+1+1) = 3

Answer 4

So you find the integral of 3 over the volume given by 0<=z<=3 and x^2 +y^2 <= 1

Answer 5

In cylinderical coordinates this region is described as 0<=z<=3 , 0<=theta<=2pi and 0<=r<=1

also remember for cylinderical coordinates "dx dy" becomes "r dr dtheta".

The jacobian of the polar substitution is r

Answer 6

\[=\int\limits_{0}^{1} \int\limits_{0}^{2 \pi} \int\limits_{0}^{3} 3r dz d \theta dr = 3 [ \int\limits_{0}^{1} r dr][\int\limits_{0}^{2 \pi} d \theta] [ \int\limits_{0}^{3} dz] =3[\frac{1}{2}][2 \pi][3] = 9 \pi \]

Answer 7

if my integral is written as \[\int\limits \overrightarrow{r} \cdot \overrightarrow{dv}\] does that change any of what you said?