Multivariate Calculus by Alder

By Alder

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Multidisciplinary Methods for Analysis Optimization and Control of Complex Systems

This booklet includes lecture notes of a summer season tuition named after the overdue Jacques Louis Lions. The summer season tuition was once designed to alert either Academia and to the expanding function of multidisciplinary tools and instruments for the layout of complicated items in a variety of parts of socio-economic curiosity.

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Given R(x, y) dx ∧ dy, is there always a P dx + Q dy that gives R(x, y) = ∂Q ∂P − ? ∂x ∂y It is easy to see that there are lots of them. 1. Let ψ = xy sin(y) dx ∧ dy be a “spin field” on R2 . Is it derived from some 1-form ω = P dx + Q dy? Solution: Yes, put P = 0 and Q = x2 y sin(y)/2 Then ∂Q ∂P − = xy sin(y) ∂x ∂y All I did was to set P to zero and integrate Q with respect to x. This is a bit too easy to be interesting. It stops being so silly if we do it on R3 , as we shall see later. 2. It should be obvious that just as we had the derivative taking 0-forms to 1-forms, so we have a process for getting 2-forms from 1-forms.

We used an actual parametrisation only to make it easier to evaluate it. 1. For any continuous vector field F on Rn and any differentiable curve c, the value of the integral of F over c does not depend on the choice of parametrisation of c. I shall prove this soon. 1. For the vector field F on R2 given by F x y −y x evaluate the integral of F along the straight line joining 1 0 to 0 . 3. INDEPENDENCE OF PARAMETRISATION 37 Parametrisation 1 c : [0, 1] −→ R2 1 0 (1 − t) t 0 1 +t 1−t t = c = −1 1 Then the integral is −t 1−t t=1 t=0 q −1 1 dt 1 t + 1 − tdt = = 1dt = t]10 = 1 0 This looks reasonable enough.

X1   ..   Vn  .   xn  Rn to R, and P V= P V+ c 0 V P Where I have specified the endpoints only since V has the independence of path properly. For every point P ∈ D I define ϕ ( P ) to be above equation as P 0 V, and I can rewrite the P V ϕ(P ) = ϕ(P ) + P   1 = ϕ(P ) + t=0     V1 (c(t))    .. q   .   Vn (c(t)) 1 0 0 .. 5. CLOSED LOOPS AND CONSERVATISM    where c(t) =   x1 − a + t x2 .. 45     for 0 ≤ t ≤ a.  xn Since the integration is just along the x1 line we can write   x1  x2    x=x1  x3  ϕ(P ) = ϕ(P ) + V1   dx  ..

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