34 1. Differential Equations and Their Solutions Exercise 1–22. Show that a differentiable function F is a con- stant of the motion if and only if its directional derivative at any point x in the direction V (x) is zero, i.e., k ∂F (x) ∂x k Vk(x) = 0. The flow is called isometric (or distance preserving) if for all points p, q in Rn and all times t, φt(p) φt(q) = p q , and it is called volume preserving if for all open sets O of Rn, the volume of φt(O) equals the volume of O. Given a linear map B : Rn Rn, we get a bilinear map ˆ : Rn × Rn R by ˆ(u, v) = Bu, v , where Bu, v is just the inner product (or dot product) of Bu and v. We say that the flow preserves the bilinear form ˆ if ˆ((Dφ t )x(u), (Dφt)x(v)) = ˆ(u, v) for all u, v in Rn and all x in Rn. Here, the linear map D(φt)x : Rn Rn is the differential of φt at x i.e., if the components of φt(x) are Φi(x, t), then the matrix of D(φt)x is just the Jacobian matrix ∂Φi(x,t) ∂xj . We will denote the de- terminant of this latter matrix (the Jacobian determinant) by J(x, t). We note that because φ0(x) = x, ∂Φi(x,0) ∂xj is the identity matrix, and it follows that J(x, 0) = 1. Exercise 1–23. Since, by definition, t φt(x) is a solution of dx dt = V (x), ∂Φi(x,t) ∂t = Vi(φt(x)). Using this, deduce that ∂t ∂Φi(x,t) ∂xj = k ∂Vi(φt(x)) ∂xk ∂Φk(x,t) ∂xj and in particular that ( ∂t ) t=0 ∂Φi(x,t) ∂xj = ∂Vi(x) ∂xj . Exercise 1–24. We define a scalar function div(V ), the diver- gence of V , by div(V ) := i ∂Vi ∂xi) . Using the formula for the derivative of a determinant, show that ( ∂t t=0 J(x, t) = div(V )(x). Exercise 1–25. Now, using the “change of variable formula” for an n-dimensional integral, you should be able to show that the flow generated by V is volume preserving if and only if div(V ) is identically zero. Hint: You will need to use the group property, φt+s = φt φs. Exercise 1–26. Let Bij denote the matrix of the linear map B. Show that a flow preserves ˆ if and only if k Bik ∂Vk ∂xj + ∂Vk ∂xi Bkj = 0. Show that the flow is isometric if and only if it preserves ˆ (i.e.,
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