Mathematical Physics Vol 1

Chapter 4. Field theory

188

Solution

− 8 π .

Exercise 131 Calculate the itegral

I c ( x 2 − 2 xy ) d x +( x 2 y + 3 ) d y , if the boundary of the region is defined by the intersection of the lines y 2 = 2 x and x = 2: a) directly, b) using Green’s theorem.

Solution

128 / 5.

Exercise 132 Calulate the integral

Z c ( 6 xy − y 2 ) d x +( 3 x 2 − 2 xy ) d y , along the cycloid c , defined by x = θ − sin θ , y = 1 − cos θ , from point A ( 0 , 0 ) topoint B ( π , 2 ) .

Solution

6 π 2 − 4 π .

Exercise 133 Show that the surface

A = x R

d x d y ,

after the transformation x = x ( u , v ) and y = y ( u , v ) , is given by

∂ x ∂ u ∂ x ∂ v

∂ y ∂ u ∂ y ∂ v

A = x R

∂ ( x , y ) ∂ ( u , v ) ≡

J d u d v ,

gde je J =

,

J - Jacobian of the transformation.