296 Chapter 5 Higher Dimensions and Other Coordinates
(a) (b)
Figure 2 (a) Distributed forces. (b) Concentrated forces.Figure 3 Forces on a piece of membrane.(a) (b)
Figure 4 Forces (a) in thexu-plane; (b) in theyu-plane.Looking at projection on thexu-andyu-planes (Figs. 4a, 4b), we see that
the sum of forces in thex-direction isσ y(cos(β)−cos(α)),andthesumof
forces in they-direction is x(cos(δ)−cos(γ )). It is desirable that both these
sums be zero or at least negligible. Therefore we shall assume thatα,β,γ,and
δare all small angles. Because we know that
tan(α)=∂u
∂x, tan(γ )=∂u
∂yand so forth, when the derivatives are evaluated at some appropriate point near
(x,y), we are assuming that the slopes∂u/∂xand∂u/∂yof the membrane are
very small.