Summary of Results from Previous Courses Grad, Div, Curl and the Laplacian in Cartesian Coordinates ∂ ∂ ∂ In Cartesian coordinates, ∇ = ∂x , ∂y , ∂z . For a scalar field Φ(x) and a vector field F(x) = (F1 , F2 , F3 ), we define: ∂Φ ∂Φ ∂Φ Gradient ∇Φ = , , (“grad Phi”) ∂x ∂y ∂z Divergence Curl Laplacian
∂F2 ∂F3 ∂F1 + + ∂x ∂y ∂z ∂F3 ∂F2 ∂F1 ∂F3 ∂F2 ∂F1 ∇×F= − , − , − ∂y ∂z ∂z ∂x ∂x ∂y
∇. F =
∂2Φ ∂2Φ ∂2Φ ∇ Φ = ∇ . (∇Φ) = + + ∂x2 ∂y 2 ∂z 2 2
(“div F”) (“curl F”) (“del-squared Phi”)
The normal to a surface Φ(x) = constant is parallel to ∇Φ.
Grad, Div and the Laplacian in Polar Coordinates Cylindrical Polars (r, θ, z) When the components (F1 , F2 , F3 ) of F are measured in cylindrical polar coordinates, ∂Φ 1 ∂Φ ∂Φ ∇Φ = , , ∂r r ∂θ ∂z 1 ∂ 1 ∂F2 ∂F3 (rF1 ) + + r ∂r r ∂θ ∂z 1 ∂ ∂Φ 1 ∂2Φ ∂2Φ r + 2 2 + ∇2 Φ = r ∂r ∂r r ∂θ ∂z 2
∇. F =
Note: the formulae for plane polar coordinates (r, θ) are obtained by setting
∂ ∂z
= 0.
Spherical Polars (r, θ, φ) When the components (F1 , F2 , F3 ) of F are measured in spherical polar coordinates, ∂Φ 1 ∂Φ 1 ∂Φ ∇Φ = , , ∂r r ∂θ r sin θ ∂φ 1 ∂ 2 1 ∂ 1 ∂F3 (r F ) + (F sin θ) + 1 2 r2 ∂r r sin θ ∂θ r sin θ ∂φ 1 ∂ 1 ∂ ∂Φ 1 ∂2Φ 2 2 ∂Φ ∇ Φ= 2 r + 2 sin θ + 2 2 r ∂r ∂r r sin θ ∂θ ∂θ r sin θ ∂φ2
∇. F =
Mathematical Methods II Natural Sciences Tripos Part IB