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Multi-variable Calculus
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Notion
Multi-variable Calculus
Differentials
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$df = f_xdx + f_ydy+f_zdz$ο»Ώ
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$df \neq \Delta f$ο»Ώ
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$\Delta f = f_x\Delta x + f_y\Delta y + f_z\Delta z$ο»Ώ
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$df$ο»Ώ is used to encode infinitesimal changes
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used to act as a placegolder value
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divide wrt time to get rate of change $\rightarrow$ο»Ώ CHAIN RULE
Chain Rule with More Variables
Let $w = f(x, y)$ο»Ώ when $x(u, v), y(u, v)$ο»Ώ 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: $\vec{\nabla}w \ \perp \text{ level surfaces}$ο»Ώ (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 $\vec{\nabla}w$ο»Ώ is the direction of fastest increase of $w$ο»Ώ
Goal: minima/maximize a multi-variable function ($min/max\ \ f(x, y, z)$ο»Ώ) where $x, y, z$ο»Ώ are not independent and $\exists$ο»Ώ $g(x, y, z) = c$ο»Ώ.
$\vec{\nabla}f = \lambda \vec{\nabla}g$
Basic idea: to find $(x, y)$ο»Ώ where the level curves of $f$ο»Ώ and $g$ο»Ώ are tangent to each other ($\vec{\nabla}f \parallel \vec{\nabla}g$ο»Ώ).