MATH1231 Tutorial Summaries

Tutor: Peter O’Brien

  1. Surface z = F(x,y)
    • Level curves: $F(x,y) = c, c = z$ (basically let z be constant and plot x and y at each point)
  2. Partial Derivatives
(1)
\begin{align} \frac {∂F}{∂x}, \frac {∂F}{∂y} \end{align}
(2)
\begin{align} \frac {∂^2F}{∂x^2}, \frac {∂^2F}{∂y^2}, \frac {∂^2F}{∂y∂x} \frac {∂^2F}{∂x∂y} \end{align}
  1. Tangent Planes
    • $Z = z0 + Fx(x-x0) + Fy(y-y0)$
    • Normal (one of them) = $(Fx, Fy, -1)^T$ (transverse
  2. Total Differential Approximation
(3)
\begin{align} ∆F ≈ \frac {∂F}{∂x}∆x + \frac {∂F}{∂y} ∆y \end{align}
  • Change in F is approximately equal to change in x times partial derivative of x plus change in y times partial derivative of y