Normal vector in surface integral of vector field

[foo9008]

Homework Statement

when the normal vector n is oriented upward , why the dz/dx and dz/dy is negative ? shouldn't the k = positive , while the dz/dx and dz/dy is also positive?

Homework Equations

The Attempt at a Solution

is the author wrong ? [/B]

 

[BvU]

Try a few examples and you'll see.
Remember when vectors are perpendicular
and when lines are perpendicular

 

[foo9008]

Try a few examples and you'll see.
Remember when vectors are perpendicular
and when lines are perpendicular

Can you give some example?

 

[BvU]

z = x

 

[foo9008]

z = x

i still don't understand what do you mean, can you explain further?

 

[vela]

The surface $\sigma$ is defined by $\phi(x,y,z)=0$ where $\phi(x,y,z)=z-f(x,y)$, and the normal is the gradient of $\phi(x,y,z)$. What do you get when you calculate that?

 

[foo9008]

The surface $\sigma$ is defined by $\phi(x,y,z)=0$ where $\phi(x,y,z)=z-f(x,y)$, and the normal is the gradient of $\phi(x,y,z)$. What do you get when you calculate that?

why it will become like this ?

 

[BvU]

Why don't you simply fill it in and see ? Asking 'why' forever doesn't make sense. Make a sketch for the simplest case if 4.5.1 is too complicated.

 

[vela]

why it will become like this ?

I'm sure this is covered in your textbook.