Arc Length
Consider a curve u=u(t),v=v(t) on a surface r =r (u, v )for t 0 ≤t≤t 1. The
element of arc length ds associated with this curve can be determined from the vector
element
dr =∂r
∂u
du +∂r
∂v
dv
using
ds^2 =dr ·dr =
(
∂r
∂u
du +∂r
∂v
dv
)
·
(
∂r
∂u
du +∂r
∂v
dv
)
ds^2 =E du^2 + 2F du dv +G du^2
(7 .58)
where E, F, G are given by the equations (7.54). The length of the curve is then given
by the integral
s=arc length =
∫t 1
t 0
√
E
(du
dt
) 2
+ 2F
du
dt
dv
dt
+G
(dv
dt
) 2
dt (7 .59)
where the limits of integration t 0 and t 1 correspond to the endpoints associated with
the curve as determined by the parameter t.
The Gradient, Divergence and Curl
The gradient is a field characteristic that describes the spatial rate of change of
a scalar field. Let φ=φ(x, y, z )represent a scalar field, then the gradient of φis a
vector and is written
grad φ=
∂φ
∂x ˆe^1 +
∂φ
∂y ˆe^2 +
∂φ
∂z ˆe^3. (7 .60)
Here it is assumed that the scalar field φ=φ(x, y, z)possesses first partial derivatives
throughout some region R of space in order that the gradient vector exists. The
operator
∇= ∂
∂x
ˆe 1 + ∂
∂y
ˆe 2 + ∂
∂z
ˆe 3 (7 .61)
is called the “del ”operator or nabla operator and can be used to express the gradient
in the operator form
grad φ=∇φ=
(
∂
∂x
ˆe 1 + ∂
∂y
ˆe 2 + ∂
∂z
eˆ 3
)
φ. (7 .62)
Note that the operator is not commutative and ∇φ=φ∇.
If v =v (x, y, z ) = v 1 (x, y, z )ˆe 1 +v 2 (x, y, z )ˆe 2 +v 3 (x, y, z)ˆe 3 denotes a vector field with
components which are well defined, continuous and everywhere differential, then the
divergence of v is defined
div v =
∂v 1
∂x +
∂v 2
∂y +
∂v 3
∂z (7 .63)
