What is the meaning of $d\vec S$ in a surface integral?

Question

Can someone explain if I have a surface $z= 9-x^2-y^2$

• What would $\vec{n}$ be?
• What would $d\vec{S}$ be?

Why is $d\vec{S}$ $(2x,2y,1)$ and not $(2x,2y,1)/\sqrt{4x^2+4y^2+1}$? Thanks!

Answer 1

suppose the surface is described using $z=f(x,y)$

$d\overrightarrow{S} = \hat{n}dS = \langle -f_x, -f_y, 1\rangle dxdy$

$\hat{n} = \dfrac{\langle -f_x, -f_y, 1\rangle}{||\langle -f_x, -f_y, 1\rangle||}$

$dS = ||\langle -f_x, -f_y, 1\rangle|| dxdy $

In your particular problem :

$d\overrightarrow{S} = \hat{n}dS = \langle 2x, 2y, 1\rangle dxdy$

$\hat{n} = \dfrac{\langle 2x, 2y, 1\rangle}{||\langle 2x, 2y, 1\rangle||}$

$dS = ||\langle 2x, 2y, 1\rangle|| dxdy $