Stone M.'s Mathematics for Physics. 1 PDF

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2 51 Norms and Inner Products We are often interested in “how large” a function is. This leads to the idea of normed function spaces. There are many measures of function size. Suppose R(t) is the number of inches per hour of rainfall. If your are a farmer you are probably most concerned with the total amount of rain that falls. A big rain has big |R(t)| dt. If you are the Urbana city engineer worrying about the capacity of the sewer system to cope with a downpour, you are primarily concerned with the maximum value of R(t).

E. obeys ∇ · v = 0. The second comes from varying h, and is the Bernoulli equation stating that we have P = P0 (atmospheric pressure) everywhere on the free surface. The third, from the variation of φ at y = 0, states that no fluid escapes through the lower boundary. ˙ is somewhat Obtaining and interpreting the last equation, involving h, trickier. It comes from the variation of φ on the upper boundary. The variation of S due to δφ is δS = ρ0 ∂ ∂ δφ + ∂t ∂x δφ ∂φ ∂x + ∂ ∂y δφ ∂φ ∂y − δφ ∇2 φ dtdxdy.

So 2(x, y) − λ(1, 1) = 0, ⇒ (x, y) = λ (1, 1) 2 x + y = 1, ⇒ λ = 1, =⇒ 1 1 (x, y) = ( , ). 154) where < ei > denotes the space spanned by the vectors ei and < ei >⊥ is the its orthogonal complement. Thus ∇f lies in the space spanned by the vectors ∇gi , so there must exist n numbers λi such that n ∇f = i=1 λi ∇gi . 155) The numbers λi are called Lagrange multipliers. 156) 36 CHAPTER 1. CALCULUS OF VARIATIONS with the undetermined multipliers λi subsequently being fixed by imposing the requirement that gi = 0.

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Mathematics for Physics. 1 by Stone M.

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