Lectures 1, 2 and 3 Flashcards
What is the total differential?
Can write f(x,y) as df = df/dx *dx + df/dy *dy
How do you work out if a PDE is exact or inexact?
Compare it with the total differential expression, and see if you can find an expression for f(x,y) (if not then is inexact, vice versa)
How do you test if an equation is exact? When is it inexact?
Take the total differential, differentiate both the differentials by the opposite variable (x or y), and if these are equal to eachother, it is exact. (if not the inexact).
i.e. d/dy(df/dx) and d/dx(df/dy)
How do you get one of Maxwell’s thermodynamic relationships from the thermodynamic potential?
Sub in the total differential and find what needs to be equal for it to be exact.
What is the chain rule equation?
df/dt = df/dy * dy/dt + df/dx * dx/dt
Define x and y in polar coordinates.
x = ρ*cos(θ) y = ρ*sin(θ)
Write the chain rule for polar coordinates.
dg/dρ = df/dx * dx/dρ + df/dy * dy/dρ,
dg/dθ = df/dx * dx/dθ + df/dy * dy/dθ
Then substitute in the transformations for x and y differentiated to get a term without rho or theta.
How can we write the change of coordinate systems in terms of a sum?
df/duj = sum over i of df/dxi * dxi/duj, where we use x1,x2,x3…xi instead of x,y,z, and u1, u2….uj instead of rho, theta, phi etc.
How can the differential of dx/dρ be confusing? Why does this happen?
dx/dρ and dρ/dx both equal cos(θ). This happens as we need to be explicit about which variables are held constant.
What is the reciprocity relationship and when does it hold?
(dy/dx) at z = (dx/dy)^-1 at z, only if the variable being held constant is the same on both sides of the equation.
What does ∇Ф give you?
The maximum gradient vector at any point.
What is the directional derivative?
The derivative of our function, Ф, along any direction, u, given by:
∇Ф . u(hat)
