Question

Q 4.106)

94 CHAPTER 4 Gradient. DIVEI CHAPTER 4 HO Prove V x (Ax B) BV, A)-(A-ViB +AV B B- VA ANSWERS TO 4.jP1 Prove VIA B)B- VIA +(A- ViB + Bx (VxA)+AxIVB. 442 10-4 4.102 Show that A(by 42+d-i+ (3-vk s imotational Find d sach that A-Vd 443, 5.71-i 4.j00 Show that E-r/e is imotational Find & sach that E-y6 and such that dea)0 where a>0 444. (al-4 44040 Suppose A and B are imotational Prove that A x B is solenoidal 4.45. 3 447, -2 4.1a Suppose fir) is differentiable. Prove that firir is irotational 4.48 r/3 L106) Is there a differentiable vector function V soch that ta) curl Vr. tb)curl V2++ 3k? 1f so, find V 4.49. 4.107. Show that solutions to Matwell's eqgsations 4.50 (2-r v-H-, V E-4sp Vx H- V xE- where p is a function of , , and , and c is the velocity of light, assamed constant, are piven by 451 I aA H-Vx A E-V 452. where A and d called the rector and scalar potentials, respectively, satisty the equations 4.58 (21+ 4.59 (254

Answer 1

It is not possible to have any vector function whose curl is taken then we get these vectors

As if take gradient of both the sides then as gradient of a curl is zero so left hand side is 0

And we can see that gradient of r is 3

So R.H.S. is not equal to L.H.S

Hence there is no vector function V

Similarly gradient of 2 i + j + 3k is 6

So for the vectors gradient is not 0

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