22 1 Mathematical Physics
1.5 (a) If the field is centrally represented byF=f(x,y,z),r=f(r)r, then it
is conservative conditioned by curlF=0, that is the field is irrotational.
(b) What should be the functionF(r) so that the field is solenoidal?
1.6 Evaluate∫
cA.drfrom the pointP(0,^0 ,0) toQ(1,^1 ,1) along the curve
r=ˆit+ˆjt^2 +ˆkr^3 withx=t,y=t^2 ,z=t^3 , whereA=yˆi+xzˆj+xyzˆk
1.7 Evaluate∮
cA.draround the closed curveCdefined byy=x(^2) andy (^2) = 8 x,
withA=(x+y)ˆi+(x−y)ˆj
1.8 (a) Show thatF=(2xy+z^2 )ˆi+x^2 ˆj+xyzkˆ, is a conservative force field.
(b) Find the scalar potential.
(c) Find the work done in moving a unit mass in this field from the point
(1, 0, 1) to (2, 1,−1).
1.9 Verify Green’s theorem in the plane for
∫
c(x+y)dx+(x−y)dy, whereCis
the closed curve of the region bonded byy=x^2 andy^2 = 8 x.1.10 Show that
∫
sA.ds =12
5 πR(^2) , whereS is the sphere of radiusR and
A=ixˆ^3 +ˆjy^3 +kzˆ^3
1.11 Evaluate
∫
rA.draround the circlex(^2) +y (^2) =R (^2) in thexy-plane, where
A= 2 yˆi− 3 xˆj+zˆk
1.12 (a) Prove that the curl of gradient is zero.
(b) Prove that the divergence of a curl is zero.
1.13 Ifφ=x^2 y− 2 xz^3 , then:
(a) Find the Gradient.
(b) Find the Laplacian.
1.14 (a) Find a unit vector normal to the surfacex^2 y+xz = 3 at the point
(1,−1, 1).
(b) Find the directional derivative ofφ=x^2 yz+ 2 xz^3 at (1, 1,−1) in the
direction 2ˆi− 2 ˆj+kˆ.
1.15 Show that the divergence of an inverse square force is zero.
1.16 Find the angle between the surfacesx^2 +y^2 +z^2 =1 andz=x^2 +y^2 −1at
the point (1,+1,−1).
1.2.2 FourierSeriesandFourierTransforms.................
1.17 Develop the Fourier series expansion for the saw-tooth (Ramp) wavef(x)=
x/L,−L<x<L, as in Fig. 1.2.
1.18 Find the Fourier series of the periodic function defined by:
f(x)= 0 ,if−π≤x≤ 0
f(x)=π,if 0≤x≤π