Mathematical Tools for Physics
7—Operators and Matrices 193 question: Do I start over with a new basis, or can I use the work that I’ve already done to transfo ...
7—Operators and Matrices 194 and I pulled the standard trick of changing the last dummy label of summation fromitojso that I can ...
7—Operators and Matrices 195 which determines the first column of the matrix(S), then~e′ 2 determines the second column. (S) = ( ...
7—Operators and Matrices 196 For the example Eq. ( 29 ) the eigenvectors are given in Eq. ( 30 ). In components these are ~e′ 1 ...
7—Operators and Matrices 197 Take the matrix ( 1 2 0 1 ) You can’t diagonalize this. If you try the standard procedure, here is ...
7—Operators and Matrices 198 Write this as a pair of equations, using the velocity as an independent variable. dx dt =vx and dvx ...
7—Operators and Matrices 199 Pick the initial conditions thatx(0) = 0andvx(0) =v 0. You must choosesomeinitial conditions in ord ...
7—Operators and Matrices 200 But this assumes that you already know the ratings of the sites, and that’s what you’re trying to f ...
7—Operators and Matrices 201 Problems 7.1 Draw a picture of the effect of these linear transformations on the unit square with v ...
7—Operators and Matrices 202 7.6 You have a mass attached to four springs in a plane and that are in turn attached to four walls ...
7—Operators and Matrices 203 7.13 A cube of uniform volume mass density, massm, and sideahas one corner at the origin of the coo ...
7—Operators and Matrices 204 7.21 Interpret~σ.A~asσxAx+σyAy+σzAzand prove that ~σ.A~σ~ .B~=A~.B~+i~σ.A~×B~ where the first term ...
7—Operators and Matrices 205 7.26 The set of Hermite polynomials starts out as H 0 = 1, H 1 = 2x, H 2 = 4x^2 − 2 , H 3 = 8x^3 − ...
7—Operators and Matrices 206 7.31 The characteristic polynomial of a matrixMisdet(M−λI). Iis the identity matrix andλis the vari ...
7—Operators and Matrices 207 7.39 The commutator of two matrices is defined to be[A,B] =AB−BA. Show that this commutator satisfi ...
Multivariable Calculus The world is not one-dimensional, and calculus doesn’t stop with a single independent variable. The ideas ...
8—Multivariable Calculus 209 The problem is that thenotationis ambiguous. When I write∂f/∂xit doesn’t tell be what I’m to hold c ...
8—Multivariable Calculus 210 and read (sort of) that the infinitesimal change in the functionfis the slope times the infinitesim ...
8—Multivariable Calculus 211 For example, take the functionf(x,y) =x^2 +y^2. At the point(x,y) = (1,2), the differential is df(1 ...
8—Multivariable Calculus 212 this Leibnitz notation is designed to lead you in the right direction, so it’s not an accident that ...
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