The Laplacian of a scalar field "f" can be interpreted as the divergence of the gradient of f. However, the Laplacian of a vector field has no such interpretation because you can't take the gradient of a vector field. In order to make sense of the "gradient" of a vector field, we introduce the concept of a "projection operator", which generalizes into the concept of a "Dyadic" [Transparancy 1].
A Dyadic identity operator is defined [Transparancy 2]. It is shown how to interpret the "gradient" of a vector field in terms of a dyadic with 9 components. Expanding the Laplacian of a vector field, in this Dyadic manner, shows that the Dyadic interpretation is consistent with our previous expression for the components of the Laplacian of a vector field.
Dyadics are useful for obtaining Taylor Polynomials in 3-dimensions [Transparancies 3 and 4]. The expansion of a scalar field about a point, in the direction of a given unit vector, can be expressed in terms of the directional derivative, which we previously showed could be written in terms of the gradient. This leads to a double-del Dyadic in the second order Taylor polynomial [Transparancy 4]. This is defined to be the "Hessian," which can be written in matrix form, as a matrix of second derivatives [Transparancy 5].
A simple example of a second-order Taylor expansion in 3 dimensions was given [Transparancy 6], in which the Hessian matrix is mostly zeros. (This won't be true in general.)
A second example was considered [Transparancy 7]. In this one, the Taylor expansion was found about two different points, both being points at which the gradient is zero, and the Hessian matrix is diagonal. However, only one point turned out to be extremal (a local maximum), whereas the other point turned out to be a saddle point [Transparancy 8]. This led us to the "negative semi-definite condition" which the Hessian must satisfy in order for a function to have a local maximum. This is a generalization of the second derivative test.
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