Mathematical Tools for Physics
11—Numerical Analysis 353 11.37 Derive Eq. ( 49 ). (b) Explain why the plausibility arguments that follow it actually say someth ...
Tensors You can’t walk across a room without using a tensor (the pressure tensor). You can’t balance the wheels on your car with ...
12—Tensors 355 in this cased~will be mostly in thexdirection (k 1 ) and is not aligned withF~. In any case there is a relation b ...
12—Tensors 356 x^2 +y^2 =R^2 is arelationbetween X and Y, buty= √ R^2 −x^2 is afunction. The domain of a function is the set of ...
12—Tensors 357 Now obviously the function defined byA~.~v, whereA~is a fixed vector, is a linear functional. The burden of this ...
12—Tensors 358 Let β= −αf(~ω 1 ) f(~ω 2 ) . With this choice then,f(α~ω 1 +β~ω 2 ) = 0, meaning that the combination is inM. But ...
12—Tensors 359 Similarly for multilinear functionals, with as many arguments as you want. Now apply the representation theorem f ...
12—Tensors 360 The conductivity tensor relates current to the electric field: ~=^11 σ ( E~ ) . In general this is not just a sc ...
12—Tensors 361 computation in an arbitrary basis, but for the moment it’s a little simpler to start out with the more common ort ...
12—Tensors 362 indices has been chosen for later convenience, with the sum on the first index of theTji. This equation isthe fun ...
12—Tensors 363 Notice how the previous choice of indices has led to the conventional result, with the first index denoting the r ...
12—Tensors 364 take the same tensor and compute its components in a different basis. Note: The method I’ll describe here is rath ...
12—Tensors 365 The two expressions ( 19 ) and ( 20 ) represent the same thing, so equate the coefficients ofˆe 1 and ofˆe 2 : T ...
12—Tensors 366 As a check to be sure nothing has gone wrong, it’s easy to compute the effect of^11 Ton the original vectors ˆxan ...
12—Tensors 367 But, ~u. 11 T(~v) =uieˆi.vj^11 T(~ei) =uivjˆei.(Tkjˆek) =uivjTij. The last step comes from the orthonormality of ...
12—Tensors 368 The generalization of this statement to an arbitrary rotation should now be obvious. If the components of the ten ...
12—Tensors 369 for all vectors~uand~v. You should see that using the same symbol,T, for both functions doesn’t cause any trouble ...
12—Tensors 370 Theithcomponent of which is Tjivj If you write this as a square matrix times a column matrix, the only difference ...
12—Tensors 371 Many of the common tensors such as the tensor of inertia and the dielectric tensor are symmetric. The magnetic fi ...
12—Tensors 372 Proof: Consider the functionΛ−αΛ′whereαis a scalar. Pick any three independent vectors~v 10 ,~v 20 , ~v 30 as lon ...
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