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Generalizing the Derivative

The number of different ideas, whether from physical science or other disciplines, that can be understood with reference to the ``meaning'' of a derivative from the calculus of scalar functions is very very large. Our ideas about many topics, such as price elasticity, strain, stability, and optimization, are connected to our understanding of a derivative.

In vector calculus, there are generalizations to the derivative from basic calculus that acts on a scalar and gives another scalar back:

gradient ($\nabla$ ):

A derivative on a scalar that gives a vector.

curl ( $\nabla \times$ ):

A derivative on a vector that gives another vector.

divergence ( $\nabla \cdot$ ):

A derivative on a vector that gives scalar.

Each of these have ``meanings'' that can be applied to a broad class of problems.

The gradient operation on $f(\vec{x}) = f(x,y,z) = f(x_1 , x_2 , x_3 )$ ,

$$f = \ensuremath{\nabla}f \left( \frac{\partial f}{\partial x}, \f... ...artial x}, \frac{\partial }{\partial y}, \frac{\partial }{\partial z} \right) f$$ (13-1)

has been discussed previously. The curl and divergence will be discussed below.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-13)

(html Lecture-13)

(xml+mathml Lecture-13)

Gradient of a several $1/r$ potentials