Chapter 1
Mathematical Physics
1.1 Basic Concepts and Formulae
Vector calculus
Angle between two vectors, cosθ=|AA||.BB|
Condition for coplanarity of vectors,A.B×C= 0Del∇=∂
∂xˆi+∂
∂yˆj+ ∂
∂zkˆGradient∇φ=(
∂φ
∂xˆi+∂φ
∂yˆj+∂φ
∂zkˆ)
Divergence
If V(x,y,z) = V 1 ˆi+ V 2 ˆj +V 3 kˆ, be a differentiable vector field, then
∇.V=∂∂xV 1 +∂∂yV 2 +∂∂zV 3Laplacian∇^2 =
∂^2
∂x^2+
∂^2
∂y^2+
∂^2
∂z^2(Cartesian coordinatesx,y,z)∇^2 =
1
r^2∂
∂r(
r^2∂
∂r)
+
1
r^2 sinθ∂
∂θ(
sinθ∂
∂θ)
+
1
r^2 sin^2 θ∂^2
∂Φ^2
(Spherical coordinatesr,θ,Φ)∇^2 =∂^2
∂r^2+
1
r∂
∂r+
1
r^2∂^2
∂θ^2+
∂^2
∂z^2(Cylindrical coordinatesr,θ,z)Line integrals(a)
∫
Cφdr1