Understanding Flux of Vector Fields: Equations, Solutions, and Common Mistakes

[yecko]

Homework Statement

Example 2:[/B]

Homework Equations

Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y)

The Attempt at a Solution

Why do my attempt is wrong? The example is using the foundational formula while I use the stock formula from the book, why is there a negative sign difference between the answers? Or is that my formula used inappropiately?

Thanks![/B]

 

[LCKurtz]

One reason PF discourages the use of images is that they are difficult to edit. On your first line you have the equation $g = y = x^2$, whatever that means. You are likely using the formula for a surface of the form $z = g(x,y)$. The surface $y = x^2$ is not that kind of surface because $y$ and $x$ are not independent. The easiest way to represent the surface is $y = g(x,z)$. In any case, however you did it, your normal vector is in the wrong direction. The $y$ component of your normal vector must be negative.

 

[yecko]

The y coordinate of your normal vector must be negative.

Thanks for pointing out the problem...

Flux=integrate -Pgx-Qgy+R of the proj. area on xy plane for z=g(x,y)

but how can we see the direction of normal vector in this formula?
and how to correct it? (simply by adding a negative sign?)

 

[LCKurtz]

Thanks for pointing out the problem...

but how can we see the direction of normal vector in this formula?
and how to correct it? (simply by adding a negative sign?)

I can't tell how you got your normal or what formula you used because you didn't show your work. What I would do is parameterize the surface like this$$
\vec R(x,z) = \langle x, x^2, z\rangle$$and get a normal by $\vec R_x\times \vec R_z$ and take it or its opposite, whichever has a negative $y$ component.

 

[yecko]

Alright! I believe I've got it! thanks!