This book will not use a great deal of vector or multivariate calculus,
but a *little* general familiarity with it will greatly help the
student with e.g. multiple integrals or the idea of the force being the
negative gradient of the potential energy. We will content ourselves
with a few definitions and examples.

The first definition is that of the *partial derivative*. Given a
function of many variables
, the partial derivative of the
function with respect to (say)
is written:

(147) |

and is just the regular derivative of the

In many problems, the variables *are* independent and the partial
derivative is equal to the regular derivative:

(148) |

In other problems, the variable
might *depend* on the variable
. So might
. In that case we can form the *total* derivative
of
with respect to
by including the variation of
caused by
the variation of the other variables as well (basically using the chain
rule and composition):

(149) |

Note the different

There are several ways to form vector derivatives of functions,
especially *vector* functions. We begin by defining the *gradient* operator, the basic vector differential form:

(150) |

This operator can be applied to a scalar multivariate function to form its gradient:

(151) |

The gradient of a function has a magnitude equal to its

If we wish to take the vector derivative of a vector function there are
two common ways to go about it. Suppose
is a vector function
of the spatial coordinates. We can form its *divergence*:

(152) |

or its

(153) |

These operations are extremely important in physics courses, especially the more advanced study of electromagnetics, where they are part of the differential formulation of Maxwell's equations, but we will not use them in a required way in this course. We'll introduce and discuss them and work a rare problem or two, just enough to get the