2.5 Vector Analysis

Basic Operations

∇=(∂()/∂(x)∂()/∂(y)∂()/∂(z))

Coordinates

Cartesian

𝔽=((F_x),(F_y),(F_z)), d(V)=d(x)*d(y)*d(z)

∇f=((f_x),(f_y),(f_z))

Cylindrical

(r,θ,z), x=r*cos(θ), y=r*sin(θ)

∇ƒ=((f_r),1/r,(f_θ),(f_z)) d(V)=r*d(r) d(θ)*d(z)

∇⋅𝔽=1/r*(∂_r)(r*(F_r))+1/r*(∂_θ)((F_θ))+(∂_z)((F_z))

Spherical

(ρ,θ,φ), x=ρ*sin(φ)*cos(θ)

∇ƒ=((f_ρ),1/ρ*(f_θ),1/ρ*sin(θ)*(f_φ))

d(V)=ρ2*sin(θ)*d*ρ*d*θ*d*φ

∇⋅𝔽=1/(ρ2)*(∂_ρ)((ρ2*(F_P)))+1/(ρ*sin(θ))*(∂_θ)(sin(θ)*(F_θ))+1/(ρ*sin(θ))*(∂_φ)((F_Φ))

Integral Theorems

Useful Identities

∇×(∇ƒ)=0

∇⋅(∇×𝔽)=0

∇⋅(ƒ*𝔽)=ƒ∇⋅𝔽+∇ƒ⋅𝔽

∇×(ƒ*𝔽)=∇ƒ×𝔽+ƒ∇×𝔽

∇×(∇×𝔽)=∇(∇⋅𝔽)-Δ*𝔽

Vector Identities

A×(B×C)=B*(A⋅C)-C*(A⋅B)

∇(ƒ*g)=ƒ∇g+g∇ƒ

More Cheat Sheets ->