Mathematical Physics Vol 1

Chapter 4. Field theory

158

Figure 4.27: Movement of a planet around the Sun.

where θ is the angle between p and r 0 , and p is the magnitude of vector p . Given that r · ( v × h )= r × v · h = h · h = h 2 , we have h 2 = γ Mr + rp cos θ and h 2 / γ M 1 +( p / γ M ) cos θ . This equation represents the equation of the ellipse with respect to the polar cylindrical coordinate system, with its origin at the center of the ellipse. r = h 2 γ M + p cos θ =

Exercise 97 If v =( 3 x 2 + 6 y ) i − 14 yz j + 20 xz 2 k , determine R c A ( 1 , 1 , 1 ) along the curve c determined by a) parametric equations x = t , y = t 2 , z = t 3 ,

v · d r frompoint O ( 0 , 0 , 0 ) to point

b) the line OBCA , composed of segments OB , BC and CA , where the coordinates of the points on the line are as follows: O ( 0 , 0 , 0 ) , B ( 1 , 0 , 0 ) , C ( 1 , 1 , 0 ) , A ( 1 , 1 , 1 ) , c) a straight line passing through points O ( 0 , 0 , 0 ) and A ( 1 , 1 , 1 ) .

Solution For the given vector field v the following stands Z c v · d r = Z c [( 3 x 2 + 6 y ) i − 14 yz j + 20 xz 2 k ] · ( d x i + d y j + d z k ) = Z c ( 3 x 2 + 6 y ) d x − 14 yz d y + 20 xz 2 d z . a) Parameter values t = 0and t = 1 correspond to the points O ( 0 , 0 , 0 ) and A ( 1 , 1 , 1 )