By George Turrell (Auth.)

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**Extra resources for Mathematics for Chemistry & Physics**

**Sample text**

Clearly, the addition of A to B yields the same result (see Fig. 2); hence. A^B=B (2) ^A=R and vector addition is commutative. When three vectors A, B and C are added, the resultant R is the diagonal of the parallelepiped whose edges are the vectors, as shown in Fig. 3. The same result is obtained if any two of the vectors are combined and the sum is added to the third. Thus, (A+B)-\-C =A-]-(B -\-C) = {C -\-A)-\-B =A^B Fig. 2 The vector sum R =A-\-B. -\^C =R, (3) 65 4. VECTOR ANALYSIS Fig. 3 The vector sum R =A-\-B -{-C.

And, as A^- approaches Aa in the limit, —2 2 2 ds = dx -\-dy . (24) Then the differential ds becomes the hypotenuse of the triangle shown in Fig. 9. The same result is obtained if s decreases as x increases. 8 THE MEAN-VALUE THEOREM AND L'HOSPITAL'S RULE* An important theorem, often attributed to Lagrange,^ can be written in the form f(b) - f(a) f(x\) = 7 . (25) b—a Here f\xi) is the derivative dy/dx evaluated at a point x\ which is intermediate with respect to points a and b. From Fig. 10 it should be evident that there is always some point x = xi where the slope of the curve is equal to the right-hand side of Eq.

11, where it is compared with the twoand three-term approximations derived from Eq. (40). At values of x near zero these approximations become increasingly accurate. 1050. This development is often employed in computer programs. Clearly, for a given value of x the number of terms used is determined by the precision required in the numerical result. * It can be summarized as follows for a series defined by Eq. (32). (i) If limn-^^\Cn+\/Cn\ < 1 the series converges absolutely,^ and thus converges.