= (a) There is an important theorem in vector calculus that says that YĞ = 0 (where Ğ is some differentiable vector func

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A There Is An Important Theorem In Vector Calculus That Says That Yg 0 Where G Is Some Differentiable Vector Func 1 (130.09 KiB) Viewed 44 times

= (a) There is an important theorem in vector calculus that says that YĞ = 0 (where Ğ is some differentiable vector function) implies and is implied by Ğ = V x H (where H is another differentiable function). To prove this we note first of all that G = 7 x ħ implies that Y. G = 0 (see problem 3-7). To show that Y. G = 0 implies that we can write G=, the simplest procedure is to give ť: Х = : Hx = 0, 2 = у TO H.-G..,,,a) de' -- Gule'n , a) dx + [ Glow.a)av [(2,9da° ry Hz = 2' X (, ', zdy', хо = where Xo and Yo are arbitrary constants. Show by direct calculation that if VG = 0, then Ğ = Ý x H. (b) Is the vector function à specified in (a) unique? That is, can we alter it in any way without invalidating the relation G = ï ~ ? =