Schwere, Elektricität und Magnetismus:386

Bernhard Riemann: Schwere, Elektricität und Magnetismus
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VECTOR ANALYSIS.

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. Differetntial 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)

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

(3)

$d\lbrack \alpha \times \beta \rbrack =d\alpha \times \beta +\alpha \times d\beta ,\,$

(4)

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

(5)

$d\lbrack (\alpha \cdot \beta )\gamma \rbrack =(d\alpha \cdot \beta )\gamma +(\alpha \cdot d\beta )\gamma +(\alpha \cdot \beta )d\gamma .\,$

(6)

     46. Differential coeffcient with respect to a scalar.—The quotient obtained by dividing the differential of a vector due to the variation of any scalar of which it is a function by the differential of that scalar is called the differential coefficient of the vector with respect to the scalar, and is indicated in the same manner as the differential coefficients of ordinary analysis.