Mathematical Tools for Physics
6—Vector Spaces 153 there is an upper bound (many in fact) and we take the smallest of these as the norm. On−∞< x <+∞, the ...
6—Vector Spaces 154 9 Scalar products and norms used here are just like those used for example 5. The difference is that the sum ...
6—Vector Spaces 155 6.7 Bases and Scalar Products When there is a scalar product, a most useful type of basis is the orthonormal ...
6—Vector Spaces 156 on the domain−L < x <+Lis just an example of Fourier series, and the components offin this basis are F ...
6—Vector Spaces 157 How much bigger than zero the left side is will depend on the parameterλ. To find the smallest value that th ...
6—Vector Spaces 158 so.) The third requirement, the triangle inequality, takes a bit of work and uses the inequality Eq. ( 19 ). ...
6—Vector Spaces 159 is the real numbers. The same thing happens with vectors when the dimension of the space is infinite — in or ...
6—Vector Spaces 160 Problems 6.1 Fourier series represents a choice of basis for functions on an interval. For suitably smooth f ...
6—Vector Spaces 161 6.6 For the vectors in three dimensions, ~v 1 =ˆx+y, ~vˆ 2 =yˆ+ˆz, ~v 3 =zˆ+ˆx use the Gram-Schmidt procedur ...
6—Vector Spaces 162 6.11 Which of these are vector spaces? (a) all polynomials of degree 3 (b) all polynomials of degree≤ 3 [Is ...
6—Vector Spaces 163 6.17 The equation describing the motion of a string that is oscillating with frequencyω about its stretched ...
6—Vector Spaces 164 (c) The vector~v′is unique. (d)(−1)~v=~v′. 6.21 For the vector space of polynomials, are the two functions{1 ...
6—Vector Spaces 165 6.26 Verify that Eq. ( 12 ) does satisfy the requirements for a scalar product. 6.27 A variation on problem ...
6—Vector Spaces 166 6.33 Show that the sequence of rational numbersan = ∑n ∑n k=1^1 /k is not a Cauchy sequence. What about k=1^ ...
6—Vector Spaces 167 6.37 Take the functionsf 1 ,f 2 , andf 3 from the preceding problem and sketch the shape of the functions r ...
Operators and Matrices You’ve been using operators for years even if you’ve never heard the term. Differentiation falls into thi ...
7—Operators and Matrices 169 f(~v) α ~v f(~v 1 +~v 2 ) α ~v^1 +~v^2 What happens if you change the argument of this function, mu ...
7—Operators and Matrices 170 Add and subtract the same thing on the right side of the equation (add zero) to get ~r×F~=~r× d~p d ...
7—Operators and Matrices 171 supporting rod is attached to the axis, then~r×~pfor the mass on the right is pointing up and to th ...
7—Operators and Matrices 172 This function satisfies the same linearity equations as Eq. ( 1 ). When you multiply~ωby a constant ...
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