Mathematical Physics Vol 1

4.6 Examples

171

Solution x 2 cos y + xz sin y = const.

Exercise 110 Let F =( x + 2 y ) i − 3 z j + x k , φ = 4 x + 3 y − 2 z , and S be the region of the surface 2 x + y + 2 z = 6 bounded by the planes x = 0, x = 1, y = 0 and y = 2. Calculate the following integrals: a) x S ( ∇ × F ) · n d S . b) x S φ n d S .

Solution a) 1, b) 2 i + j + 2 k .

Exercise 111 Let F =( 2 x 2 − 3 z ) i − 2 xy j − 4 x k , and V be a closed region bounded by the planes x = 0, y = 0, z = 0 and 2 x + 2 y + z = 4. Calculate a) y V ( ∇ · F ) d V , b) y V ( ∇ × F ) d V .

Solution a) 8 / 3, b)

8 3

( j − k ) .

Exercise 112 Calculate the integral

( 2 , 1 ) Z

( 10 x 4 − 2 xy 3 ) d x − 3 x 2 y 2 d y ,

( 0 , 0 )

along the path x 4 − 6 xy 3 = 4 y 2 .