Page 15-3
n independent variables
φ
(x
1
, x
2
, …,x
n
), and a vector of the functions [‘x
1
’
‘x
2
’…’x
n
’]. Function HESS returns the Hessian matrix of the function
φ
, defined
as the matrix
H
= [h
ij
] = [
∂φ
/
∂
x
i
∂
x
j
], the gradient of the function with respect to
the n-variables,
grad
f = [
∂φ
/
∂
x
1
,
∂φ
/
∂
x
2
, …
∂φ
/
∂
x
n
], and the list of
variables [‘x
1
’ ‘x
2
’…’x
n
’]. Consider as an example the function
φ
(X,Y,Z) = X
2
+ XY + XZ, we’ll apply function HESS to this scalar field in the following
example in RPN mode:
Thus, the gradient is [2X+Y+Z, X, X]. Alternatively, one can use function
DERIV as follows: DERIV(X^2+X*Y+X*Z,[X,Y,Z]), to obtain the same result.
Potential of a gradient
Given the vector field,
F
(x,y,z) = f(x,y,z)
i
+g(x,y,z)
j
+h(x,y,z)
k
, if there exists a
function
φ
(x,y,z), such that f =
∂φ
/
∂
x, g =
∂φ
/
∂
y, and h =
∂φ
/
∂
z, then
φ
(x,y,z)
is referred to as the potential function for the vector field
F
. It follows that
F
=
grad
φ
=
∇φ
.
The calculator provides function POTENTIAL, available through the command
catalog (
‚N
), to calculate the potential function of a vector field, if it
exists. For example, if
F
(x,y,z) =
xi
+ y
j
+ z
k
, applying function POTENTIAL
we find:
Since function SQ(x) represents x
2
, this results indicates that the potential
function for the vector field
F
(x,y,z) = x
i
+ y
j
+ z
k
, is
φ
(x,y,z) = (x
2
+y
2
+z
2
)/2.
Notice that the conditions for the existence of
φ
(x,y,z), namely, f =
∂φ
/
∂
x, g =
∂φ
/
∂
y, and h =
∂φ
/
∂
z, are equivalent to the conditions:
∂
f/
∂
y =
∂
g/
∂
x,
∂
f/
∂
z
=
∂
h/
∂
x, and
∂
g/
∂
z =
∂
h/
∂
y. These conditions provide a quick way to
determine if the vector field has an associated potential function. If one of the
conditions
∂
f/
∂
y =
∂
g/
∂
x,
∂
f/
∂
z =
∂
h/
∂
x,
∂
g/
∂
z =
∂
h/
∂
y, fails, a potential
