Mathematical Physics Vol 1

4.6 Examples

141

Observe the following two cases 1) when the origin of the coordinate system is inside the closed curved line and 2) when the origin of the coordinate system is outside the closed curved line, see Fig. 4.26. First case. Observe the closed curved line ABCDA (see Figure) which encircles the origin of the coordinate system. When moving along the curve, the top of the position vector starts at point A ( r = r A , ϕ 0 = 0), moves along the curve and returns to the starting point (the curve is closed !!!). Thus, the angle was increased by 2 π (0 ≤ ϕ ≤ 2 π ), and the line integral is equal to 2 π Z 0 d ϕ = 2 π .

Figure 4.26: Origin of the coordinate system: (a) inside, (b) outside the closed curved line.

Second case. For the closed curve PQRSP (see Fig. 4.26) that does not encircle the origin of the coordinate system, the angle changes from ϕ = ϕ 0 in P to ϕ = ϕ 0 after completing the full circle. The line integral is equal to ϕ 0 Z ϕ 0 d ϕ = 0 .

4.6.4 Mixed problems

Exercise 70 The following vector is given

v =( x + 2 y + az ) i +( bx − 3 y − z ) j +( 4 x + cy + 2 z ) k . a) Determine the constants a , b , c so that the vector field is potential. b) Find a scalar function φ , whose gradient is equal to vector v , for the values of constants determined under a).