# Normal vector in surface integral of vector field

1. Jul 18, 2016

### foo9008

1. The problem statement, all variables and given/known data
when the normal vector n is oriented upward , why the dz/dx and dz/dy is negative ? shouldnt the k = positive , while the dz/dx and dz/dy is also positive?

2. Relevant equations

3. The attempt at a solution
is the author wrong ?

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2. Jul 19, 2016

### BvU

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

3. Jul 19, 2016

### foo9008

Can you give some example???

4. Jul 19, 2016

### BvU

z = x

5. Jul 19, 2016

### foo9008

i still dont understand what do you mean, can you explain further?

6. Jul 20, 2016

### vela

Staff Emeritus
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?

7. Jul 20, 2016

### foo9008

why it will become like this ?

8. Jul 20, 2016

### 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.

9. Jul 24, 2016

### vela

Staff Emeritus
I'm sure this is covered in your textbook.