Chapter 1
Mathematical Physics
1.1 Basic Concepts and Formulae
Vector calculus
Angle between two vectors, cos θ =
A.B
| A||B|
Condition for coplanarity of vectors, A.B × C = 0
Del
∇ =
∂
∂ x
ˆ
i +
∂
∂ y
ˆ j +
∂
∂z
ˆ
k
Gradient
∇φ =
∂φ
∂ x
ˆ
i +
∂φ
∂ y
ˆ j +
∂φ
∂z
ˆ
k
Divergence
If V (x, y, z) = V 1 ˆ
i + V 2 ˆ j + V 3 ˆ
k, be a differentiable vector field, then
∇.V =
∂
∂ x
V 1 +
∂
∂ y
V 2 +
∂
∂z
V 3
Laplacian
∇
2
=
∂
2
∂ x 2 +
∂
2
∂ y 2 +
∂
2
∂z 2 (Cartesian coordinates x, y, z)
∇
2
=
1
r 2
∂
∂r
r
2 ∂
∂r
+
1
r 2 sin θ
∂
∂θ
sin θ
∂
∂θ
+
1
r 2 sin
2
θ
∂
2
∂Φ 2
(Spherical coordinates r, θ, Φ)
∇
2
=
∂
2
∂r 2 +
1
r
∂
∂r
+
1
r 2
∂
2
∂θ 2 +
∂
2
∂z 2 (Cylindrical coordinates r, θ, z)
Line integrals
(a)
C φ dr
1