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Notion
Multi-variable Calculus
Differentials
$df = f_xdx + f_ydy+f_zdz$﻿
$df \neq \Delta f$﻿
$\Delta f = f_x\Delta x + f_y\Delta y + f_z\Delta z$﻿
$df$﻿ is used to encode infinitesimal changes
used to act as a placegolder value
divide wrt time to get rate of change $\rightarrow$﻿ CHAIN RULE
Chain Rule with More Variables
Let $w = f(x, y)$﻿ when $x(u, v), y(u, v)$﻿ then,
$dw = f_x dx + f_y dy = (f_xx_u+ f_yy_u)du + (f_xx_v+ f_yy_v)dv = \frac{\partial f}{\partial u}du + \frac{\partial f}{\partial v}dv$
$\frac{dw}{dt} = w_x \frac{dx}{dt} + w_y \frac{dy}{dt} + w_z \frac{dz}{dt} = \vec{\nabla} w.\frac{d\vec{r}}{dt}$
Note: $\vec{\nabla}w \ \perp \text{ level surfaces}$﻿ (tangent to the level surface at any given point)
$\frac{dw}{ds}|_{\hat{u}} = \vec{\nabla}w \cdot \frac{d\vec{r}}{ds} = \vec{\nabla}w \cdot \hat{u}$
Direction of $\vec{\nabla}w$﻿ is the direction of fastest increase of $w$﻿
Goal: minima/maximize a multi-variable function ($min/max\ \ f(x, y, z)$﻿) where $x, y, z$﻿ are not independent and $\exists$﻿ $g(x, y, z) = c$﻿.
$\vec{\nabla}f = \lambda \vec{\nabla}g$
Basic idea: to find $(x, y)$﻿ where the level curves of $f$﻿ and $g$﻿ are tangent to each other ($\vec{\nabla}f \parallel \vec{\nabla}g$﻿).