By Tom M. Apostol

Quantity II of "Calculus", contained during this paintings, offers multi-variable calculus and linear algebra, with functions to differential equations and chance. quantity I, offered individually, provides one-variable calculus with an creation to linear algebra.

**Read or Download Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability PDF**

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**Additional info for Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability**

**Example text**

Y,. so that (a) and (b) are satisfied when k = r . 14) Yr+1 = Xr+l - & ad+ y Construction of orthogonal sets. The Gram-Schmidt process 23 where the scalars a,, . . , a, are to be determined. For j < r, the inner product of yI+r with yj is given by since (yi, yj) = 0 if i #j . Yi) 3 (Yj, Yi) ’ If yj = 0, then yr+i is orthogonal to yj for any choice of aj, and in this case we choose aj=O. +~ is well defined and is orthogonal to each of the earlier elements yr , . . , y, . Therefore, it is orthogonal to every element in the subspace This proves (a) when k = r + 1.

6. T(x, y) = (e5, ev) . 7. T(x, y) = (x, 1). 8. T(x,y) = (x + 1,~ + 1). 9. T(x,y) = (x -y,x +y). 10. T(x,y) =(2x -y,x +y). Do the same as above for each of Exercises 11 through 15 if the transformation T: V, --f V, is described as indicated. 11. T rotates every point through the same angle q about the origin. That is, T maps a point with polar coordinates (r, 0) onto the point with polar coordinates (r, 0 + v), where q~ is fixed. Also, T maps 0 onto itself. 12. T maps each point onto its reflection with respect to a fixed line through the origin.

JG The 2n + 1 elements vO, all,. . , pzn span a subspace S of dimension 2n + 1. The elements of S are called trigonometric polynomials. IffE C(O,27r), letf, denote the projection off on the subspace S. 23) dx. off. , n. 23) approximates ,f better than any other trigonometric polynomial in S, in the sense that the norm l\f - f,ll is as small as possible. EXAMPLE 2. Approximation of continuous functions on [- 1, I] by polynomials of degree < n. Let V = C(- 1, 1)) the space of real continuous functions on [- 1, 11, and let (f, g) = S’, f(x)&) dx.