Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/45

which subsist between one vector of one system and the three vectors of the other system. If we desire to express the similar relations which sabsist between two vectors of one system and two of the other, we may take the skew products of equations (1) with equations (2), after transposing all terms in the latter. This will afford nine equations of the type

$(i.j')k'-(i.k')j'=(k.i')j-(j.i')k.$

(7)

We may divide an equation by an indeterminate direct factor. [MS. note by author.]

CHAPTER II.

concerning the differential and integral calculus of vectors.

42. Differentials of vectors.—The differential of a vector is the geometrical difference of two values of that vector which differ infinitely little. It is itself a vector, and may make any angle with the vector differentiated. It is expressed by the same sign ($d$) as the differentials of ordinary analysis.

With reference to any fixed axes, the components of the differential of a vector are manifestly equal to the differentials of the components of the vector, i.e., if $\alpha ,\beta$, and $\gamma$ are fixed unit vectors, and

$\rho =x\alpha +y\beta +z\gamma ,$

$d\rho =dx\,\alpha +dy\,\beta +dz\,\gamma .$

43. Differential of a function of several variables.—The differential of a vector or scalar function of any number of vector or scalar variables is evidently the sum (geometrical or algebraic, according as the function is vector or scalar) of the differentials of the function due to the separate variation of the several variables.

44. Differential of a product.—The differential of a product of any kind due to the variation of a single factor is obtained by prefixing the sign of differentiation to that factor in the product. This is evidently true of differentials, since it will hold true even of finite differences.

45. From these principles we obtain the following identical equations:

$d(\alpha +\beta )=d\alpha +d\beta ,$

(1)

$d(n\alpha )=dn\,\alpha +n\,d\alpha ,$

(2)

$d(\alpha .\beta )=d\alpha .\beta +\alpha .d\beta ,$

(3)

$d[\alpha \times \beta ]=d\alpha \times \beta +\alpha \times d\beta ,$

(4)

$d(\alpha .\beta \times \gamma )=d\alpha .\beta \times \gamma +\alpha .d\beta \times \gamma +\alpha .\beta \times d\gamma ,$

(5)

$d[(\alpha .\beta )\gamma ]=(d\alpha .\beta )\gamma +(\alpha .d\beta )\gamma +(\alpha .\beta )d\gamma .$

(6)