Past Paper 1 Flashcards
what is the transpose of A_ij
- A^T_ij = A_ji
if you have u_i*A_ij, and you transpose A, what does the expression transform into
- u_iA_ij = A^T_iju_i
what is the equilibrium equation of a body with mass density ρ if the force per unit area acting on a surface is denoted by the traction vector t
- int[t dS]_dV + int[ρg dV]_V = o
- o is a position vector
given t = σn, you can write int[σn dS]. how would you rewrite this to integrate over dV
- divergence theorem
- int[σn dS] = int[∇ . σ dV]
what is a simple way of remembering the divergence theorem
- if the expression involves dS and dV, its likely divergence theorem
- remove the n, turn dS to dV and ∇ . the variable
if youre told to write down an expression for the moment equilibrium, what do you do
- cross product the LHS of the equilibrium equation by a vector x
- int[x cross t dS] + int[x cross ρg dV] = o
when working in index notation, how do you use the divergence theorem to convert, say int[x*n_i dS] to its dV form
- when converting from dS to dV, you’re differentiating by dx_i
- this is shown by removing n_i, putting the remaining expression in brackets, writing a comma next to it, then indexing it with n’s corresponding index
- in this case, int[x*n_i dS] = int[(x),_i dV]
what does ε_ijk*w_j equal
- ε_ijk*w_j = W_ik
- W_is a matrix and they have 2 indices
if given a functional J, how would you use the directional derivative to prove that the stationary point of a functional is also a minimum point for non-trivial u
- if u is a stationary point, it will be a minimum if d^2J/de^2 >= 0
- you can tell if the 2nd deriv of J will be positive if the integrand is squared
what is the equivalent of ∇v . ∇u
= ∇v . ∇u = ∇.(v∇u) - v∇^2(u)
when finding the directional derivative and equating it to 0 to find the u that minimises J, what are you equating to 0 specifically
- the expression within the integral wrt dV
- not including the dS ones
say you want to minimise a functional F subject to a constraint G. how would you find the values of the variables of F to do this
- you are trying to solve ∇(F + λG) = 0
- λ = lagrange multiplier
- set H = F + λG and find stationary points when the (directional) derivative of H = 0 wrt all relevant variables including λ
- you should get a set of simultaneous equations to solve
what is the equivalent of ∇.(uv)
- ∇.(uv) = ∇u.v + u∇.v
how do you find the weak form of a differential equation
- by taking the directional derivative
- no need to equate to 0, just solve for D(u)[v]
how do you find the strong form of a differential equation
- you find the weak form first to get D(u)[v]
- then you use integration by parts to remove any derivatives of not the main variable
- then you equate the int [dV] to 0 to get the strong form
what is important to remember about how to do the integration by parts
- the ‘uv’ in the integration by parts formula is what turns into the dS integral
- you just multiply the uv by .n and integrate wrt dS
if you want to impose the boundary condition ∇u.n = h on part of boundary V denoted by S_h (on a functional J), how would you modify J to satisfy this conition?
- you subtract an extra integral int[h*v dS] integrated over S_h from J
what do the minimsation questions mean when they ask you to write the boundary conditions (I think) (I was right)
- theyre referring to the equality for the matching integrals to = 0
- so if you group all dS and dV integrals separately, after doing the derivative and parts, the bit inside the integral = 0 is the condition
using the divergence theorem, what does int[∇^2u dV] equal and when is this expression particularly useful
- int[∇^2u dV] = int[du/dn dS]
- when you have a target boundary condition but dont have the right term in the integral
what is a_i*b_i in vector notation (this is index notation)
- a_i*b_i = (a.b)
what is ε_ijka_ja_k and why
- ε_ijka_ja_k = (a x a)_i
- (a x a)_i =0
what is ∇ . (∇ x f) equal to
- ∇ . (∇ x f) = 0
