what is equal (double)
dxdy = dydx
what df = ….
df/dx dx + df/dy dy
taylor series and remainder term?
f(x+h, y+h) = f(x,y) + hfx(x, y) + kfy(x,y) + h²/2 fxx(x,y)….
R/h² + k² –> 0 as k, h –> 0
chain rule?
df/dx = df/du du/dx + df/dv dv/dx
chain rule for d²y/dx²?
d²f/du² (du/dx)² + 2d²f/dudv (du/dv) (dv/dx) + d²f/dv² (dv/dx)²
cyclic relations equal?
(dy/dx)z(dx/dz)y(dz/dy)x = -1
prove cyclic relations
write out dy = (dy/dx)z dx + (dy/dz)x dz
write out dx as well, compare with dx by putting it as LHS
how to solve Pdx + Qdy when exact
dP/dy = dQ/dx
df/dx = P ⇒ f = ∫Pdx + g(y)
⇒ df/dy = [d/dy ∫Pdx] + g’(y) = Q
find g(y) and g’(y)
∫Pdx + g(y) = const c
how to solve Pdx + Qdy when nonexact
if dP/dy and dQ/dx depend only on x, try μ(x) else μ(y)
μ(x) dP/dy = μ(x) dQ/dx + μ’(x) Q
μ(x) = e^∫[(dP/dy - dQ/dx)/ Q]dx
μ(y) = e^∫[(dQ/dx - dP/dy)/ P]dy
df/dx = μP ⇒ f = ∫μPdx + g(y)
⇒ df/dy = d/dy ∫μPdx + g’(y) = μQ
4 maxwell relations and how to prove each?
(T/V)s = -(P/S)v ⇒ dU = TdS - pdV
(S/V)t = (P/T)v ⇒ F = U - TS
(T/P)s = (V/S)p ⇒ H = U + PV
(S/P)t = -(V/T)p ⇒ G = F + PV
what are the stationary points?
▽f = (df/dx, df/dy)
when 0
when maximum? minimum? saddle?
min Hxx, Hyy > 0, D > 0
max Hxx, Hyy < 0, D > 0
saddle D < 0
what do their contours look like? draw
X if saddle
normal to contour lines, towards +
hessian?
D = HxxHyy - (Hxy)²
what manipulations are allowed?
adding functions, multiplying functions, functions with changing constants, division, cancellations
how to solve conditional stationary points with lagrange multipliers
f(x,y) over constraint g(x,y) = 0
L = f - λg
dL/dx = 0
dL/dy = 0
dL/dλ = g = 0
consider f² if easier + extend for more variables and constraints
lagrange: not to forget what solutions?
± for √ and 0s
how to draw contours (3)
local max / min - tangent to contour
identify h=0 - draw lines
axes - shows where local max/min are
Ellipsoid formula
(x²/a²) + (y²/b²) + (z²/c²) = 1
