PHYS 301 -- Introduction to Mathematical Physics
Chapter 6 Vector Analysis (Section 9-10)
(i)
The Green’s theorem is like the two-dimensional version of the fundamental theorem of
calculus. It can be applied to evaluate integrals directly in some cases, although not very
frequently. However, it is used to derive the divergence theorem and Stokes’ theorem, which are
used much more frequently.
(ii)
In using these theorems, we can try to evaluate integrals in one way or the other, depends
on which is easier to do.
(iii)
If the vector field is V = ρv , where ρ is the density and v is the velocity field of the
fluid, the volume integral of ∇ ⋅ V is the amount of mass flowing out of the volume per unit
time. If there is no mass source or sink within the volume, then this amount has to be the same
the negative of the rate of change of mass within the volume. Apply this concept to an arbitrarily
€
€
∂ρ
small volume, the fluid
+ ∇ ⋅ V = 0.
€ must satisfy the continuity equation:
∂t
(iv)
This concept can be applied to other vector field, e.g. if ρ is the charge density instead,
∂ρ
we must have
+ ∇ ⋅ J = 0 , where J is the current€density.
∂t
€ the most recognized application of the
(v)
The Gauss’s law in electromagnetism is probably
divergence theorem, where the vector field is the electric field. Most often the surface integral of
€ electric field can be done easily based on symmetry, and that would relate to the volume
normal
integral of charge density (the total charge within a volume) so that the electric field for a given
charge distribution can be found.
Key equations:
(9.7)
Green’s theorem in the plane: an area integral may be expressed as a line
integral around the boundary of the area.
(10.17)
Divergence theorem: the volume integral of the divergence of a vector
field can be expressed as an integral of the normal component of the
vector over the surface enclosing the volume.
(10.9)
Equation of continuity. This not only applies to fluid, but also to many
other cases, e.g. electric current and charge density.
(10.23)-(10.24)
Guass’s law, which is the divergence theorem applied to the vector field of
electric field E. This shows that ∇ ⋅ E = ρ /ε0 is related to charge density
(one of the Maxwell equations).
Examples: 6.10.1, 6.10.5, 6.10.9
€