By Serge Lang

The current direction on calculus of numerous variables is intended as a textual content, both for one semester following a primary path in Calculus, or for a yr if the calculus series is so established. For a one-semester direction, it doesn't matter what, one may still hide the 1st 4 chapters, as much as the legislation of conservation of power, which gives a gorgeous software of the chain rule in a actual context, and ties up the math of this path with normal fabric from classes on physics. Then there are approximately probabilities: One is to hide Chapters V and VI on maxima and minima, quadratic varieties, serious issues, and Taylor's formulation. possible then end with bankruptcy IX on double integration to around off the one-term direction. the opposite is to enter curve integrals, double integration, and Green's theorem, that's Chapters VII, VIII, IX, and X, §1. This types a coherent whole.

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Remark. We take the intervals of definition for our curves to be open, closed, or also half-open or half-closed. When we define the derivative of a curve, it is understood that the interval of definition contains more than one point. e. a + h lies in the interval. If a is a left end point, the quotient is considered only for h > O. If a is a right end point the quotient is considered only for h < O. Then the usual rules for differentiation of functions are true in this greater generality, and thus Rules 1 through 4 below, and the chain rule of §2 remain true also.

Two vectors A, B are said to be parallel if there exists a number c =1= 0 such that cA = B. Two lines are said to be parallel if, given two distinct points PI' QI on the first line and P 2, Q2 on the second, the vectors and are parallel. Two planes are said to be parallel (in 3-space) if their normal vectors are parallel. They are said to be perpendicular if their normal vectors are perpendicular. The angle between two planes is defined to be the angle between their normal vectors. [I, §6] 39 PLANES Example 4.

Carry out the proofs of CP 1 through CP 4. 9. Compute the area of the parallelogram spanned by the following vectors. (a) A = (3, -2,4) and B = (5, 1, 1) (b) A = (3, 1,2) and B = (-1,2,4) (c) A = (4, -2,5) and B = (3, 1, -1) (d) A = (- 2,1,3) and B = (2, - 3,4) Do the next exercises after you have read Chapter II, §1. 10. -d-t--'----"- = X(t) x -d-t- dX(t) + -d-t- 11. Show (using only Exercise 10) that atd [X(t) x X'(t)] = X(t) x X"(t). 12. Let Y(t) = X(t)· (X'(t) x X"(t». Show that Y'(t) = X(t)· (X'(t) x X"'(t»).