Hp 50g Graphing Calculator Uživatelský manuál stáhnout pdf (Strana 473)

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Page 15-3

as the matrix H = [h

] = [∂φ/∂x

∂x

], the gradient of the function with respect to

the n-variables, grad f = [ ∂φ/∂x

, ∂φ/∂x

, … ∂φ/∂x

], and the list of

variables [‘x

’ ‘x

’…’x

’]. Consider as an example the function φ(X,Y,Z) = X

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 + yj + zk, applying function POTENTIAL we

find:

Since function SQ(x) represents x

, this results indicates that the potential

function for the vector field F(x,y,z) = xi + yj + zk, is φ(x,y,z) = (x

)/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 function

φ(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

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