Can anyone confirm this possible error in 'Vector Calculus' by Matthews?

[Syrus]

Homework Statement

Find the surface integral of u dot n over S where S is the part of the surface z = x + y² with z<0 and x>-1, u is the vector field u = (2y+x,-1,0) and n has a negative z component.

Homework Equations

In the text leading up to the end-of-chapter exercises (where this question arises), n is specified as a unit normal vector.

The Attempt at a Solution

The solution claims that ndS = (1,2y,-1)dxdy. But isn't this ignoring the fact that n should be a unit vector? Obviously, the position vector for the surface in parameter form is r = (x,y,x+y²), and n is obtained by taking the partial derivative of r with respect to y, the partial derivative of r with respect to x, and forming the cross product of the two (in that order). But then, shouldn't this resulting vector be divided by the magnitude of the cross product- which, by my calculations happens to be √(2 + 4y²), which is never 1?

 

[LCKurtz]

Homework Statement

Find the surface integral of u dot n over S where S is the part of the surface z = x + y² with z<0 and x>-1, u is the vector field u = (2y+x,-1,0) and n has a negative z component.

Homework Equations

In the text leading up to the end-of-chapter exercises (where this question arises), n is specified as a unit normal vector.

The Attempt at a Solution

The solution claims that ndS = (1,2y,-1)dxdy. But isn't this ignoring the fact that n should be a unit vector? Obviously, the position vector for the surface in parameter form is r = (x,y,x+y²), and n is obtained by taking the partial derivative of r with respect to y, the partial derivative of r with respect to x, and forming the cross product of the two (in that order). But then, shouldn't this resulting vector be divided by the magnitude of the cross product- which, by my calculations happens to be √(2 + 4y²), which is never 1?

Your surface is parameterized as $\vec R(x,y) = \langle x, y, x+y^2\rangle$. Read my post #13 in this thread:

https://www.physicsforums.com/showthread.php?t=611873

with u = x and v = y and you will see why the book is correct.

 

[Syrus]

Understood. Thank you, LCKurtz.