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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 ⋅ ∂ [ ]+ j ⋅ ∂ [ ]+ k ⋅ ∂ [ ]
∂x ∂y ∂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
Thus, the gradient is [2X+Y+Z, X, X].
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