22
1 Mathematical Physics
1.5 (a) If the field is centrally represented by F = f (x, y, z), r = f (r )r, then it
is conservative conditioned by curl F = 0, that is the field is irrotational.
(b) What should be the function F(r ) so that the field is solenoidal?
1.6 Evaluate
c A . dr from the point P(0, 0, 0) to Q(1, 1, 1) along the curve
r = ˆ
it + ˆ jt
2
+ ˆ
kr
3 with x = t, y = t
2
, z = t
3 , where A = y ˆ
i + xz ˆ j + x yz ˆ
k
1.7 Evaluate
c A . dr around the closed curve C defined by y = x
2 and y
2
= 8x,
with A = (x + y) ˆ
i + (x − y) ˆ j
1.8 (a) Show that F = (2x y + z
2 ) ˆ
i + x
2 ˆ j + x yz ˆ
k, 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, where C is
the closed curve of the region bonded by y = x
2 and y
2
= 8x.
1.10 Show that
s A . ds =
12
5
π R
2 , where S is the sphere of radius R and
A = ˆ
i x
3
+ ˆ j y
3
+ ˆ
kz
3
1.11 Evaluate
r A . dr around the circle x
2
+ y
2
= R
2 in the x y-plane, where
A = 2y ˆ
i − 3x ˆ j + z ˆ
k
1.12 (a) Prove that the curl of gradient is zero.
(b) Prove that the divergence of a curl is zero.
1.13 If φ = x
2 y − 2xz
3 , then:
(a) Find the Gradient.
(b) Find the Laplacian.
1.14 (a) Find a unit vector normal to the surface x
2 y + xz = 3 at the point
(1, −1, 1).
(b) Find the directional derivative of φ = x
2 yz + 2xz
3 at (1, 1, −1) in the
direction 2 ˆ
i − 2 ˆ j + ˆ
k.
1.15 Show that the divergence of an inverse square force is zero.
1.16 Find the angle between the surfaces x
2
+ y
2
+ z
2
= 1 and z = x
2
+ y
2
− 1 at
the point (1, +1, −1).
1.2.2 Fourier Series and Fourier Transforms
1.17 Develop the Fourier series expansion for the saw-tooth (Ramp) wave f (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
1 Mathematical Physics
1.5 (a) If the field is centrally represented by F = f (x, y, z), r = f (r )r, then it
is conservative conditioned by curl F = 0, that is the field is irrotational.
(b) What should be the function F(r ) so that the field is solenoidal?
1.6 Evaluate
c A . dr from the point P(0, 0, 0) to Q(1, 1, 1) along the curve
r = ˆ
it + ˆ jt
2
+ ˆ
kr
3 with x = t, y = t
2
, z = t
3 , where A = y ˆ
i + xz ˆ j + x yz ˆ
k
1.7 Evaluate
c A . dr around the closed curve C defined by y = x
2 and y
2
= 8x,
with A = (x + y) ˆ
i + (x − y) ˆ j
1.8 (a) Show that F = (2x y + z
2 ) ˆ
i + x
2 ˆ j + x yz ˆ
k, 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, where C is
the closed curve of the region bonded by y = x
2 and y
2
= 8x.
1.10 Show that
s A . ds =
12
5
π R
2 , where S is the sphere of radius R and
A = ˆ
i x
3
+ ˆ j y
3
+ ˆ
kz
3
1.11 Evaluate
r A . dr around the circle x
2
+ y
2
= R
2 in the x y-plane, where
A = 2y ˆ
i − 3x ˆ j + z ˆ
k
1.12 (a) Prove that the curl of gradient is zero.
(b) Prove that the divergence of a curl is zero.
1.13 If φ = x
2 y − 2xz
3 , then:
(a) Find the Gradient.
(b) Find the Laplacian.
1.14 (a) Find a unit vector normal to the surface x
2 y + xz = 3 at the point
(1, −1, 1).
(b) Find the directional derivative of φ = x
2 yz + 2xz
3 at (1, 1, −1) in the
direction 2 ˆ
i − 2 ˆ j + ˆ
k.
1.15 Show that the divergence of an inverse square force is zero.
1.16 Find the angle between the surfaces x
2
+ y
2
+ z
2
= 1 and z = x
2
+ y
2
− 1 at
the point (1, +1, −1).
1.2.2 Fourier Series and Fourier Transforms
1.17 Develop the Fourier series expansion for the saw-tooth (Ramp) wave f (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