Mathematics · Ch 8 — Differentials and Partial Derivatives

Summary

8.8

Summary

Let $f:(a,b)\to\mathbb R$ be differentiable and $x_0\in(a,b)$ . The linear approximation $L$ of $f$ at $x_0$ is $L(x)=f(x_0)+f'(x_0)(x-x_0)$ for all $x\in(a,b)$ .
Absolute error $=$ Actual value $-$ Approximate value. $\ \text{Relative error}=\dfrac{\text{Absolute error}}{\text{Actual value}}$ . $\ \text{Percentage error}=\text{Relative error}\times100=\dfrac{\text{Absolute error}}{\text{Actual value}}\times100$ .
For $f:(a,b)\to\mathbb R$ differentiable, $x\in(a,b)$ , and $\Delta x$ the increment given to $x$ , the differential of $f$ is $df=f'(x)\,\Delta x$ .
All the limit theorems (limit rules) for functions of one variable also hold true for functions of several variables.
Let $A=\{(x,y)\mid a<x<b,\,c<y<d\}\subset\mathbb R^2$ $A = {(x, y) ∣ a < x < b, c < y < d} \subset R^{2}$ , $F:A\to\mathbb R$ $F : A \to R$ , $(x_0,y_0)\in A$ $(x_{0}, y_{0}) \in A$ .
- (i) $F$ has a partial derivative with respect to $x$ at $(x_0,y_0)$ if $\displaystyle\lim_{h\to0}\dfrac{F(x_0+h,y_0)-F(x_0,y_0)}{h}$ exists, denoted $\dfrac{\partial F}{\partial x}(x_0,y_0)$ ; similarly with respect to $y$ using $k\to0$ , denoted $\dfrac{\partial F}{\partial y}(x_0,y_0)$ .
- Clairaut's Theorem: if $F_{xy}$ and $F_{yx}$ exist on $A$ and are continuous on $A$ , then $F_{xy}=F_{yx}$ on $A$ .
- $u:A\to\mathbb R$ is harmonic in $A$ if $\dfrac{\partial^2u}{\partial x^2}+\dfrac{\partial^2u}{\partial y^2}=0$ for all $(x,y)\in A$ (Laplace's equation).
- (i) The linear approximation of $F$ at $(x_0,y_0)$ is $F(x,y)=F(x_0,y_0)+\dfrac{\partial F}{\partial x}(x_0,y_0)(x-x_0)+\dfrac{\partial F}{\partial y}(x_0,y_0)(y-y_0)$ .
- (ii) The differential of $F$ is $dF=\dfrac{\partial F}{\partial x}\,dx+\dfrac{\partial F}{\partial y}\,dy$ , where $\Delta x=dx$ and $\Delta y=dy$ .
If $w$ is a function of two variables $x,y$ where $x,y$ are functions of a single variable $t$ , then $\dfrac{dw}{dt}=\dfrac{\partial w}{\partial x}\cdot\dfrac{dx}{dt}+\dfrac{\partial w}{\partial y}\cdot\dfrac{dy}{dt}$ . …

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