CHAPTER IV.
(Supplementary to Chapter II.)
concerning the differential and integral calculus of vectors.
169. If
is a vector having continuously varying values in space, and
the vector determining the position of a point, we may set
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and regard
as a function of
or of
and
Then,
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that is,
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If we set
![{\displaystyle \nabla \omega =i{\frac {d\omega }{dx}}+j{\frac {d\omega }{dy}}+k{\frac {d\omega }{dz}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ed994481ef3c2450f85d94d90060360e1a0b030)
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Here
stands for
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exactly as in No. 52, except that it is here applied to a vector and produces a dyadic, while in the former case it was applied to a scalar and produced a vector. The dyadic
represents the nine differential coefficients of the three components of w with respect to
and
just as the vector
(where
is a scalar function of
) represents the three differential coefficients of the scalar
with respect to
and
It is evident that the expressions
and
already defined (No. 54) are equivalent to
and
160. An important case is that in which the vector operated on is of the form
We have then
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where
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This dyadic, which is evidently self-conjugate, represents the six