Graphics Reference
In-Depth Information
-A-
Background
A.1 Vector Calculus
The following three differential operators are fundamental to vector calcu-
lus: the gradient
∇
, the divergence
∇·
, and the curl
∇×
. They occasionally
are written in equations as grad, div, and curl instead.
A.1.1 Gradient
The gradient simply takes all the spatial partial derivatives of the function,
returning a vector. In two dimensions,
f
(
x, y
)=
∂f
,
∂f
∂y
∇
∂x
,
and in three dimensions,
f
(
x, y, z
)=
∂f
.
∂f
∂y
,
∂f
∂z
∇
∂x
,
It can sometimes be helpful to think of the gradient operator as a symbolic
vector, e.g., in three dimensions:
=
∂
.
∂
∂y
,
∂
∂z
∇
∂x
,
The gradient is often used to approximate a function locally:
f
(
x
+Δ
x
)
≈
f
(
x
)+
∇
f
(
x
)
·
Δ
x.
In a related vein we can evaluate the
directional derivative
of the function;
that is how fast the function is changing when looking along a particular
vector direction, using the gradient. For example, if the direction is
n
,
∂f
∂n
=
∇
f
·
n.
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