In this lecture, we'll see how to write an automatic di erenti-ation engine | a program that builds the computation graph and applies the backprop updates, all without us having to calculate any …

In this lecture we will cover some very basic material in symplectic geometry. First, we will discuss classical Hamiltonian mechanics, reinterpret it in our symplectic setting and show that it was all …

Proposition 1.3. A mapping f : Cm ! Cn is pseudo-holomorphic if and only if the Cauchy-Riemann equations are satis ed, that is, writing f(z1; : : : zm) = (f1; : : : ; fn) = (u1 + iv1; : : : un + ivn); and zj = xj …

Appendix A Vector Algebra As is natural, our Aerospace Structures will be described in a Euclidean th. onal space R 3. 1 Vectors vector is used to represent quantities that have bot. magnitude and …

@H@vj = @vj@xk @H @xk two terms comes from the fact that = 0 and yj = @H @xk @vj . This proves

Consider first the incompressible NS equations (we will be using index notation) @vj = 0 @xj ⇢ @vi @vi @ ij + vj = @t @xj @xj Averaging the equations, using that the fluctuations have zero mean, yields …

Example 16 presents a solution of the Navier-Stokes equations, starting from a solution of the Stokes equations. Modify it so that it starts from zero velocity and pressure, ramping up the boundary …