Flux of a Vector Through a Surface: x²+y²=8, z-x²-y²=0

[cristina89]

Homework Statement

The velocity of a fluid is given, in cartesian coordinates (x, y, z), by \vec{v} = v_{0}x\hat{z} being a constant with velocity dimensions.
a) Calculate the flux of this vector through the closed surface composed by z-x²-y²=0 and by the plan z=8 limited by the circle with radius a=\sqrt{8} (x²+y²=8).

Homework Equations

\int\int{\vec{v}}\cdot\vec{n}\cdot dS

The Attempt at a Solution

Well. I've calculated this and it gave me 0 as a result... But I don't know if I did it right.
I've started with

\int\int{\vec{v}}\cdot\vec{n}\cdot dS = \int\int{v_{0}x}\cdot dS

So I've transformed into polar coordinates and it gave me
\int\int{v_{0}rcosθ}\cdot rdrdθ = \int\int{v_{0}rcosθ}\cdot rdrdθ
= v_{0} \int^{2\pi}_{0}\int^{8}_{0}r²cosθdrdθ

And developing this, it gave me 0 as a result. Is this correct?

 

[tiny-tim]

hi cristina89!

\int\int{\vec{v}}\cdot\vec{n}\cdot dS = \int\int{v_{0}x}\cdot dS

no, \int\int{v_{0}x}(\hat{z}\cdot\hat{n})dS

 

[cristina89]

hi cristina89!


no, \int\int{v_{0}x}(\hat{z}\cdot\hat{n})dS

ahh ok. so \int\int{v_{0}x}(\hat{z}\cdot\hat{n})dS = \int\int v_{0}x(\hat{z})\frac{v_{0}x\hat{z}}{\sqrt{v_{0}^{2}x^{2}}} right?
What should I do after that? I'm now confused if there is this normal vector in this integral. Is this correct?

 

[tiny-tim]

\int\int{v_{0}x}(\hat{z}\cdot\hat{n})dS = \int\int v_{0}x(\hat{z})\frac{v_{0}x\hat{z}}{\sqrt{v_{0}^{2}x^{2}}}

i don't understand that

(and that fraction is just \hat{z})

\hat{z}\cdot\hat{n} is cos of the slope of the normal (= sin of the slope of the surface)

 

[cristina89]

i don't understand that

(and that fraction is just \hat{z})

\hat{z}\cdot\hat{n} is cos of the slope of the normal (= sin of the slope of the surface)

Can't I think that \hat{n} = \frac{v}{|v|} = \frac{v_{0}x\hat{z}}{\sqrt{v_{0}^{2}x^{2}}} = \hat{z}?
And then \int\int{v_{0}x\hat{z}\hat{z}dS}?

Can you explain to me this "\hat{z}\cdot\hat{n} is cos of the slope of the normal (= sin of the slope of the surface)"? :S I can't understand that...

 

[HallsofIvy]

There are two surfaces involved here: the paraboloid z= x^2+ y^2 and the plane z= 8. Obviously (I hope it is obvious!) the outward normal to the plane z= 8 is \vec{k} and the inward normal is -\vec{k}.

The paraboloid z= x^2+ y^2/ can be written in vector form, using x and y as parameters, as \vec{r}(x,y)= x\vec{i}+ y\vec{j}+ z\vec{k}= x\vec{i}+ y\vec{j}+ (x^2+ y^2)\vec{l}. The two derivative vectors, with respect to x and y are
\vec{r}_x= \vec{i}+2x\vec{k} and \vec{r}_y= \vec{j}+ 2y\vec{k} are tangent vectors to the surface at each point. Their cross product,
\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 0 & 2x \\ 0 & 1 & 2y\end{array}\right|= -2x\vec{i}- 2y\vec{j}+\vec{k}
is normal to the surface and \vec{n}dS= (-2x\vec{i}- 2y\vec{j}+ \vec{k})dxdy
That, with x and y components negative, pointing inward, is the "inward normal". Multiplying by -1 to get 2x\vec{i}+ 2y\vec{j}- \vec{k} gives the "outward normal". For this problem, where you are calculating the "flux", the flux all the way through, it doesn't matter whether you use "inward" or "outward" normals as long as you are consistent, using the same for both paraboloid and plane.

 

[cristina89]

There are two surfaces involved here: the paraboloid z= x^2+ y^2 and the plane z= 8. Obviously (I hope it is obvious!) the outward normal to the plane z= 8 is \vec{k} and the inward normal is -\vec{k}.

The paraboloid z= x^2+ y^2/ can be written in vector form, using x and y as parameters, as \vec{r}(x,y)= x\vec{i}+ y\vec{j}+ z\vec{k}= x\vec{i}+ y\vec{j}+ (x^2+ y^2)\vec{l}. The two derivative vectors, with respect to x and y are
\vec{r}_x= \vec{i}+2x\vec{k} and \vec{r}_y= \vec{j}+ 2y\vec{k} are tangent vectors to the surface at each point. Their cross product,
\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 1 & 0 & 2x \\ 0 & 1 & 2y\end{array}\right|= -2x\vec{i}- 2y\vec{j}+\vec{k}
is normal to the surface and \vec{n}dS= (-2x\vec{i}- 2y\vec{j}+ \vec{k})dxdy
That, with x and y components negative, pointing inward, is the "inward normal". Multiplying by -1 to get 2x\vec{i}+ 2y\vec{j}- \vec{k} gives the "outward normal". For this problem, where you are calculating the "flux", the flux all the way through, it doesn't matter whether you use "inward" or "outward" normals as long as you are consistent, using the same for both paraboloid and plane.

Thank you!
Is it right to bound the integrals like this \int^{2∏}_{0}\int^{8}_{0}?