Potential Of A Gradient - HP 50g User Manual

as the matrix H = [h
the n-variables, grad f = [
variables ['x ' 'x
1
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 =
referred to as the potential function for the vector field F. It follows that F = grad
= .
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 =
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 function
(x,y,z) does not exist. In such case, function POTENTIAL returns an error
message. For example, the vector field F(x,y,z) = (x+y)i + (x-y+z)j + xzk, does
] = [ / x
ij i
/ x
'...'x ']. Consider as an example the function (X,Y,Z) = X
2 n
/ z, are equivalent to the conditions: f/ y = g/ x, f/ z =
x ], the gradient of the function with respect to
j
, / x , ...
1 2
/ x, g = / y, and h =
2
, this results indicates that the potential
/ x ], and the list of
n
/ z, then (x,y,z) is
2
2
+
2 2
+y +z )/2.
/ x, g =
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