General Formula for n dS
Suppose S is a surface parametrized by x and y and N is any vector normal to S (not
N
necessarily unit length). Then n dS =
dx dy. Here n is the upward unit normal.
N·k
Example: for the sphere x2 + y 2 + z 2 = a2 with N = (x, y, z), find n dS.
N
x y
Answer: n dS =
dx dy = ( , , 1) dx dy.
N·k
z z
2
(Just like if we wrote z = a − x2 − y 2 , n dS = (−zx , −zy , 1) dx dy.)
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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