By E. J. McShane

ISBN-10: 0080874266

ISBN-13: 9780080874265

ISBN-10: 0124862608

ISBN-13: 9780124862609

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Extra resources for Unified Integration (Pure and Applied Mathematics (Academic Pr))

Example text

Consider a vector a with tail at the origin and head having the Cartesian coordinates (a1 , . . , an ). We can represent this vector as the sum of n rectangular component vectors, a = a1 e1 + · · · + an en . ) Another notation for the vector a is a1 , . . , an . By the Pythagorean theorem, the magnitude of the vector a is |a| = a21 + · · · + a2n . 4: Components of a vector. 2 The Kronecker Delta and Einstein Summation Convention The Kronecker Delta tensor is defined δij = 1 if i = j, 0 if i = j.

Let f (x) be differentiable and f (ξ) = 0. • If f (x) changes sign from positive to negative as we pass through x = ξ then the point is a relative maxima. • If f (x) changes sign from negative to positive as we pass through x = ξ then the point is a relative minima. • If f (x) is not identically zero in a neighborhood of x = ξ and it does not change sign as we pass through the point then x = ξ is not a relative extrema. 1 Consider y = x2 and the point x = 0. The function is differentiable. The derivative, y = 2x, vanishes at x = 0.

They are evident from geometric considerations. a+b=b+a αa = aα commutative laws (a + b) + c = a + (b + c) α(βa) = (αβ)a associative laws α(a + b) = αa + αb (α + β)a = αa + βa distributive laws Zero and Unit Vectors. The additive identity element for vectors is the zero vector or null vector. This is a vector of magnitude zero which is denoted as 0. A unit vector is a vector of magnitude one. If a is nonzero then a/|a| is a unit vector in the direction of a. Unit vectors are ˆ. often denoted with a caret over-line, n Rectangular Unit Vectors.