MA2322 Exercises 5; 2015
10 March 2015
1. A differential form is called closed if it is in the kernel of the differential d. Let ω and θ
be closed forms. Show that ω ∧ θ is closed.
2. A differential form is called exact if it is in the image of the differential d. Let ω be an
exact form and θ be a closed form. Show that ω ∧ θ is exact.
3. let ∗ be the Hodge star operator on an oriented manifold with metric tensor. Let f be a
scalar field. The scalar field ∗d ∗ df is called the Laplacian of f .
Calculate the Laplacian of f :
(a) with respect to the usual line element on R3 and the usual orientation using the
usual coordinates
(b) with respect to the usual line element on R3 and the usual orientation using spherical
polar coordinates
(c) with respect to the line element:
ds2 = (dθ)2 + (sin θdφ)2 .
on the sphere
4. Let X be a four dimensional oriented manifold with line-element of type +, +, +, −. Let
F be a 2-form and J be a 3-form related by the Maxwell equations:
dF = 0
d ∗ F = 4πJ
Express the Maxwell equations in terms of the 3-dimensional Euclidean grad, div and
curl when the line element is:
ds2 = (dx)2 + (dy)2 + (dz)2 − (cdt)2
and
and
~ ∧ cdt + B.
~
~ dr
~ dS
F = E.
~ ∧ cdt + ρdV.
J = −~j.dS
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