The Differential and Taylor Polynomials

Differential

#todo update the Taylor series page with this

$$\text{d}f(x,y,\dots )=f_x^{\prime }dx+f_y^{\prime }dy+\dots$$

$x,y,dx,dy$ are all variables

Higher Order

Single-var:

$${\text{d}}^{n}f(x)=f^{(n)}(x)(dx{)}^n$$

Multi-var:

$${\text{d}}^{n}f(x,y,\dots )=(\text{d}f(x,y,\dots ){)}^n$$

2nd Order Two-var (case of multi-var)

$${\text{d}}^{2}f(x,y)=f_{xx}^{\prime \prime }(dx{)}^2+f_{yy}^{\prime \prime }(dy{)}^2+2f_{xy}^{\prime \prime }(dx)(dy)$$

Binomial expansion for nth order two-var

$$\begin{gathered} {\text{d}}^{n}f(x,y) \\ =\sum _{k=0}^n\frac{n!}{k!(n-k)!}\frac{{\partial }^{n}f}{\partial x^k\partial y^{n-k}}(dx{)}^k(dy{)}^{n-k} \end{gathered}$$

Geometric Significance

Tells you how much the $z$ coordinate is changing along the tangent plane

$$f(x_0+dx,y_0+dy)\simeq f(x_0,y_0)+\text{d}f(x_0,y_0)$$

Taylor Polynomials

(Taylor Series)

$$T_n(x_0+dx,y_0+dy)=f(x_0,y_0)+\text{d}f(x_0,y_0)+\dots +\frac{1}{n!}{\text{d}}^{n}f(x_0,y_0)$$

Recurrence Relation:

$$T_{n+1}(x_0+dx,y_0+dy)=T_n(x_0+dx,y_0+dy)+\frac{1}{(n+1)!}{\text{d}}^{n+1}f(x_0,y_0)$$

Note: when taking a Taylor polynomial of a polynomial function, truncate to the degree of the approximation instead of actually going through all the steps

Shortcut

1. $f(x,y,z)\to f(u)$
2. Find power series of $f(u)$
3. Plug $u$ in
4. Truncate to the desired degree

Error

• Since we’re approximating, we don’t really need to be that accurate with the error
• So instead of doing the full Lagrange maximizing thing we can just take the next derivative at the point
• Error
  • $E_n^{est}=\frac{|{\text{d}}^{n+1}f({\vec{r}}_0)|}{(n+1)!}$
• Relative Error
  • ${ϵ}_n^{est}=\frac{E_n^{est}}{|T_n({\vec{r}}_0+d\vec{r})|}\cdot 100\%$
• Linear Approximation (expanded)
  • $E_1^{est}=\frac{1}{2}[f_{xx}^{\prime \prime }(x_0,y_0)(dx{)}^2+f_{yy}^{\prime \prime }(x_0,y_0)(dy{)}^2+2f_{xy}^{\prime \prime }(x_0,y_0)(dx)(dy)]$