Potential Of A Gradient, - HP F2226A - 48GII Graphic Calculator User Manual

Graphing calculator.

n independent variables (x
'x
'...'x
']. Function HESS returns the Hessian matrix of the function , defined
2
n
as the matrix H = [h
] = [ / x
ij
the n-variables, grad f = [
variables ['x
' 'x
'...'x
']. Consider as an example the function (X,Y,Z) = X
1
2
n
+ 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 (x,y,z), such that f =
is referred to as the potential function for the vector field F. It follows that F =
=
.
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 + yj + zk, applying function POTENTIAL
we find:
Since function SQ(x) represents x
function for the vector field F(x,y,z) = xi + yj + zk, is (x,y,z) = (x
Notice that the conditions for the existence of (x,y,z), namely, f =
/ y, and h =
/ z, are equivalent to the conditions:
= 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
, x
, ...,x
), and a vector of the functions ['x
1
2
n
x
], the gradient of the function with respect to
i
j
/ x
,
/ x
, ...
1
2
/ x, g =
/ y, and h =
2
, this results indicates that the potential
'
1
/ x
], and the list of
n
2
/ z, then (x,y,z)
2
2
2
+y
+z
)/2.
/ x, g =
f/ y = g/ x, f/ z
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