Mathematical Tools for Physics - Department of Physics - University

6—Vector Spaces 134 This becomes ∣ ∣〈~u,~v〉 ∣ ∣^2 ≤〈~u,~u〉〈~v,~v〉 (6.24) This isn’t quite the result needed, because Eq. (6.19) ...

6—Vector Spaces 135 These are quotients of integers, but the limit is √ 2 and that’snot* a rational number. Within the confines ...

6—Vector Spaces 136 Exercises 1 Determine if these are vector spaces with the usual rules for addition and multiplication by sca ...

6—Vector Spaces 137 Problems 6.1 Fourier series represents a choice of basis for functions on an interval. For suitably smooth f ...

6—Vector Spaces 138 and use the Gram-Schmidt procedure to construct an orthonormal basis starting from~v 0. Compare these result ...

6—Vector Spaces 139 6.15 Modify the example number 2 of section6.3so thatf 3 =f 1 +f 2 meansf 3 (x) =f 1 (x−a) + f 2 (x−b)for fi ...

6—Vector Spaces 140 (b) The number 0 times any vector is the zero vector: 0 ~v=O~. (c) The vector~v′is unique. (d)(−1)~v=~v′. 6. ...

6—Vector Spaces 141 6.29 Do the same calculation as in problem6.7, but use the scalar product 〈 f,g 〉 = ∫ 1 0 x^2 dxf*(x)g(x) 6. ...

6—Vector Spaces 142 Now pick up the samef 1 and rotate it by 90 ◦clockwise about the positivex-axis, again finally expressing th ...

Operators and Matrices . You’ve been using operators for years even if you’ve never heard the term. Differentiation falls into t ...

7—Operators and Matrices 144 Another example of the type of function that I’ll examine is from physics instead of mathematics. A ...

7—Operators and Matrices 145 Now make this quantitative and apply it to a general rigid body. There are two basic pieces to the ...

7—Operators and Matrices 146 P~is the electric dipole moment density andE~is the applied electric field. The functionαis called ...

7—Operators and Matrices 147 could go beyond 3, and the vectors that you’re dealing with may not be the usual geometric arrows. ...

7—Operators and Matrices 148   u 1 u 2 u 3  =   f 11 f 12 f 13 f 21 f 22 f 23 f 31 f 32 f 33     v 1 v 2 v 3   is u ...

7—Operators and Matrices 149 I(~ei) = ∑ k Iki~ek I(~e 1 ) = ∫ dm [ ~e 1 (x^2 +y^2 +z^2 )−(x~e 1 +y~e 2 +z~e 3 )(x) ] =I 11 ~e 1 ...

7—Operators and Matrices 150 Don’t count on all such results factoring so nicely. In this basis, the angular velocity~ωhas just ...

7—Operators and Matrices 151 Put this in words and it says that the tensor of inertia about any point is equal to the tensor of ...

7—Operators and Matrices 152 Why is this called the parallel axis theorem when you’re translating a point (the origin) and not a ...

7—Operators and Matrices 153 The computation ofh 12 from Eq. (7.26) is   h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33  =   ...