- #1

DottZakapa

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- Homework Statement
- Consider the vector field F (x, y, z) = (0, z, y) and the surface Σ= (x,y,z)∈R^3 : x=2y^2z^2, 0≤y≤2, 0≤z≤1

oriented so that its normal vector forms an acute angle with the fundamental versor of the x–axis. Compute the flux of F through Σ.

- Relevant Equations
- flux of F through sigma

Given

##F (x, y, z) = (0, z, y)## and the surface ## \Sigma = (x,y,z)∈R^3 : x=2 y^2 z^2, 0≤y≤2, 0≤z≤1##

i have parametrised as follows

##\begin{cases}

x=2u^2v^2\\

y=u\\

z=v\\

\end{cases}##

now I find the normal vector in the following way

##\begin{vmatrix}

i & j & k \\

\frac {\partial x} {\partial u} & \frac {\partial y} {\partial u}& \frac {\partial z} {\partial u} \\

\frac {\partial x} {\partial v} & \frac {\partial y} {\partial v}& \frac {\partial z} {\partial v} \\

\end{vmatrix} =

\begin{vmatrix}

i & j & k \\

4uv^2 & 1 & 0 \\

4u^2v & 0 & 1\\

\end{vmatrix} = \vec i(1)-\vec j(4uv^2)+\vec k(- 4u^2v) ##

##\Rightarrow N(u,v) = (1,-4uv^2,- 4u^2v) ##

Is there anything wrong on the normal vectors signs? what having an acute angle with x translates in?

I don't understand why in the solution the second and third components have negative sign.

##F (x, y, z) = (0, z, y)## and the surface ## \Sigma = (x,y,z)∈R^3 : x=2 y^2 z^2, 0≤y≤2, 0≤z≤1##

i have parametrised as follows

##\begin{cases}

x=2u^2v^2\\

y=u\\

z=v\\

\end{cases}##

now I find the normal vector in the following way

##\begin{vmatrix}

i & j & k \\

\frac {\partial x} {\partial u} & \frac {\partial y} {\partial u}& \frac {\partial z} {\partial u} \\

\frac {\partial x} {\partial v} & \frac {\partial y} {\partial v}& \frac {\partial z} {\partial v} \\

\end{vmatrix} =

\begin{vmatrix}

i & j & k \\

4uv^2 & 1 & 0 \\

4u^2v & 0 & 1\\

\end{vmatrix} = \vec i(1)-\vec j(4uv^2)+\vec k(- 4u^2v) ##

##\Rightarrow N(u,v) = (1,-4uv^2,- 4u^2v) ##

Is there anything wrong on the normal vectors signs? what having an acute angle with x translates in?

I don't understand why in the solution the second and third components have negative sign.

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