HP 48gII Vector Analysis Applications, The del operator, Chapter, grad, grad, φ = ∇φ, Gradient, ∇[ ] = i ⋅ ∂∂x [ ]+ j ⋅ ∂∂y [ ]+ k ⋅ ∂∂z [ ]

Chapter 13

Vector Analysis Applications

This chapter describes the use of functions HESS, DIV, and CURL, for calculating operations of vector analysis.

The del operator

The following operator, referred to as the ‘del’ or ‘nabla’ operator, is a vector- based operator that can be applied to a scalar or vector function:

∇[ ] = i ⋅ ∂∂x [ ]+ j ⋅ ∂∂y [ ]+ k ⋅ ∂∂z [ ]

When applied to a scalar function we can obtain the gradient of the function, and when applied to a vector function we can obtain the divergence and the curl of that function. A combination of gradient and divergence produces the Laplacian of a scalar function.

Gradient

The gradient of a scalar function φ(x,y,z) is a vector function defined by gradφ = ∇φ . Function HESS can be used to obtain the gradient of a function.. The function takes as input a function of n independent variables φ(x1, x2, …,xn), and a vector of the functions [‘x1’ ‘x2’…’xn’]. The function returns the Hessian matrix of the function, H = [hij] = [∂φ/∂xi∂xj], the gradient of the function with respect to the n-variables, grad f = [ ∂φ/∂x1 ∂φ/∂x2 … ∂φ/∂xn], and the list of variables [‘x1’, ‘x2’,…,’xn’]. This function is easier to visualize in the RPN mode. Consider as an example the function φ(X,Y,Z) = X2 + XY + XZ, we’ll apply function HESS to this scalar field in the following example:

Thus, the gradient is [2X+Y+Z, X, X].

Page 13-1