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V ^ = — divbj. 54) the equa- t l S0 n 2 2 ν φ = ρ, V b = -f. 55), a matter which we consider in the next section: [see p . 64 and example (ii) on p . 66]. E x a m p l e s , (i) F i n d t h e v a l u e of k for w h i c h t h e line integral t a k e n a l o n g a c u r v e C d e p e n d s o n l y o n t h e coordinates of t h e e n d p o i n t s of C. F o r this v a l u e of k d e t e r m i n e t h e v a l u e of t h e integral t a k e n a l o n g a s u i t a b l e curve b e t w e e n t h e p o i n t s ( 0 , 0 ) a n d ( 1 , 1 ) .

4) Dependence of. a line-integral on the path of integration If a = gradcp, where φ is a single-valued scalar point function, then This shows that this particular line-integral is independent of the path of integration from A t o Β, which implies that φ grad φ · ds = 0 , for an arbitrary closed curve. Therefore j f curl grad φ · d S = 0 , for an arbitrary surface. Therefore curl grad φ = 0 . η [See also eqn. 30)] Therefore, if the line-integral / a · ds is indepenA dent of the path, then curia = 0.

Ssn i BD, d EA 41 e g i v e similar con- Hence Λ III) Ε ABDE Therefore T h e y- a n d z - c o m p o n e n t s are o b t a i n e d b y cyclic p e r m u t a t i o n . (3) Solution of grad ψ — 0 This is a 'differential equation' of vector analysis. If ψ has the same value everywhere, then grad ψ = 0 . Conversely, if grad φ = 0 , then 99 is constant. (4) Dependence of. a line-integral on the path of integration If a = gradcp, where φ is a single-valued scalar point function, then This shows that this particular line-integral is independent of the path of integration from A t o Β, which implies that φ grad φ · ds = 0 , for an arbitrary closed curve.

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