G Some Tools of Vector Analysis
667
where ω is a k-dimensional differential form and ∂∂ is a closed piecewisecontinuous boundary of the manifold whose interior contains no singularities or
holes (i.e. ‘simply connected’), and is also piecewise continuous.
Maxwell’s equations for electromagnetism and Einstein’s equations for gravitational curvature also have simple expressions in terms of differential forms. Because
the relations are not explicitly dependent on a choice of coordinates, they can
aid in finding physical solutions which also should not depend on our choice of
coordinates. 6
G.8 Some Useful Vector-Calculus Identities
∇ · (A × B) = (∇ × A) · B − (∇ × B) · A
(G.42)
∇ × (∇ × A) = −∇
2 A + ∇(∇ · A)
(G.43)
∇ × (A × B) = (B · ∇)A − (A · ∇)B + (∇ · B)A − (∇ · A)B
(G.44)
These identities are easily proved in coordinate index form, taking advantage of
Eqs. (G.12) for the cross product and (G.22) for a double cross.
G.9 Applications of Vector Calculus
The gradient, divergence, and curl of a vector field finds wide application: In
mechanics, the curl of the force vanishes if the action of that force conserves energy.
In fluid mechanics, twice the curl of the velocity field of the fluid determines the
local angular velocity of the fluid where the curl is non-zero. The curl of the velocity
field is also called the local vorticity of the fluid. Maxwell’s electrodynamics were
originally expressed in term of the divergence and curl of the electric and magnetic
fields.
G.10 Poisson’s Equation
Poisson’s equation arises in many areas of mathematical physics, including the study
of gravitational fields, electrodynamics and fluid mechanics. It takes the form of a
differential equation for a field φ given a ‘driving term’ f , also sometimes called
6 A good reference is the book by Harley Flanders, Differential Forms with Application to the
Physical Sciences, [Acad Press] (1963); [Dover Books] (1989).
667
where ω is a k-dimensional differential form and ∂∂ is a closed piecewisecontinuous boundary of the manifold whose interior contains no singularities or
holes (i.e. ‘simply connected’), and is also piecewise continuous.
Maxwell’s equations for electromagnetism and Einstein’s equations for gravitational curvature also have simple expressions in terms of differential forms. Because
the relations are not explicitly dependent on a choice of coordinates, they can
aid in finding physical solutions which also should not depend on our choice of
coordinates. 6
G.8 Some Useful Vector-Calculus Identities
∇ · (A × B) = (∇ × A) · B − (∇ × B) · A
(G.42)
∇ × (∇ × A) = −∇
2 A + ∇(∇ · A)
(G.43)
∇ × (A × B) = (B · ∇)A − (A · ∇)B + (∇ · B)A − (∇ · A)B
(G.44)
These identities are easily proved in coordinate index form, taking advantage of
Eqs. (G.12) for the cross product and (G.22) for a double cross.
G.9 Applications of Vector Calculus
The gradient, divergence, and curl of a vector field finds wide application: In
mechanics, the curl of the force vanishes if the action of that force conserves energy.
In fluid mechanics, twice the curl of the velocity field of the fluid determines the
local angular velocity of the fluid where the curl is non-zero. The curl of the velocity
field is also called the local vorticity of the fluid. Maxwell’s electrodynamics were
originally expressed in term of the divergence and curl of the electric and magnetic
fields.
G.10 Poisson’s Equation
Poisson’s equation arises in many areas of mathematical physics, including the study
of gravitational fields, electrodynamics and fluid mechanics. It takes the form of a
differential equation for a field φ given a ‘driving term’ f , also sometimes called
6 A good reference is the book by Harley Flanders, Differential Forms with Application to the
Physical Sciences, [Acad Press] (1963); [Dover Books] (1989).