Multi-variable Calculus

Differentials

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df=fxdx+fydy+fzdzdf = f_xdx + f_ydy+f_zdzdf=fx​dx+fy​dy+fz​dz﻿

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df≠Δfdf \neq \Delta fdf=Δf﻿

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Δf=fxΔx+fyΔy+fzΔz\Delta f =  f_x\Delta x + f_y\Delta y + f_z\Delta zΔf=fx​Δx+fy​Δy+fz​Δz﻿

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dfdfdf﻿ 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→﻿ CHAIN RULE

Chain Rule with More Variables

Let w=f(x,y)w = f(x, y)w=f(x,y)﻿ when x(u,v),y(u,v)x(u, v), y(u, v)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

Gradient Vector

\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: ∇⃗w ⊥ level surfaces\vec{\nabla}w \ \perp \text{ level surfaces}∇w ⊥ level surfaces﻿ (tangent to the level surface at any given point)

Directional Derivatives

\frac{dw}{ds}|_{\hat{u}} = \vec{\nabla}w \cdot \frac{d\vec{r}}{ds} = \vec{\nabla}w \cdot \hat{u}

Implications

Direction of ∇⃗w\vec{\nabla}w∇w﻿ is the direction of fastest increase of www﻿

Lagrange Multipliers

Goal: minima/maximize a multi-variable function (min/max  f(x,y,z)min/max\ \  f(x, y, z)min/max  f(x,y,z)﻿) where x,y,zx, y, zx,y,z﻿ are not independent and ∃\exists∃﻿ g(x,y,z)=cg(x, y, z) = cg(x,y,z)=c﻿.

These can be obtained on combining the given restraints with the following.

\vec{\nabla}f = \lambda \vec{\nabla}g

Basic idea: to find (x,y)(x, y)(x,y)﻿ where the level curves of fff﻿ and gg g﻿ are tangent to each other (∇⃗f∥∇⃗g\vec{\nabla}f \parallel \vec{\nabla}g∇f∥∇g﻿). 

Note: Take care that the point is indeed a maxima or minima as required and not just a saddle point (second derivative test won't be applicable so be vigilant).