Multivariable Calculus

Table of Contents


Chain Rule with More Variables

Let w=f(x,y)w = f(x, y)ο»Ώ when x(u,v),y(u,v)x(u, v), y(u, v)ο»Ώ then,

dw=fxdx+fydy=(fxxu+fyyu)du+(fxxv+fyyv)dv=βˆ‚fβˆ‚udu+βˆ‚fβˆ‚vdvdw = 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

dwdt=wxdxdt+wydydt+wzdzdt=βˆ‡βƒ—w.drβƒ—dt\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: βˆ‡βƒ—wΒ βŠ₯Β levelΒ surfaces\vec{\nabla}w \ \perp \text{ level surfaces}ο»Ώ (tangent to the level surface at any given point)

Directional Derivatives

dwds∣u^=βˆ‡βƒ—wβ‹…drβƒ—ds=βˆ‡βƒ—wβ‹…u^\frac{dw}{ds}|_{\hat{u}} = \vec{\nabla}w \cdot \frac{d\vec{r}}{ds} = \vec{\nabla}w \cdot \hat{u}


Direction of βˆ‡βƒ—w\vec{\nabla}wο»Ώ is the direction of fastest increase of wwο»Ώ

Lagrange Multipliers

Goal: minima/maximize a multi-variable function (min/maxΒ Β f(x,y,z)min/max\ \ f(x, y, z)ο»Ώ) where x,y,zx, y, zο»Ώ are not independent and βˆƒ\existsο»Ώ g(x,y,z)=cg(x, y, z) = cο»Ώ.

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

βˆ‡βƒ—f=Ξ»βˆ‡βƒ—g\vec{\nabla}f = \lambda \vec{\nabla}g

Basic idea: to find (x,y)(x, y)ο»Ώ where the level curves of ffο»Ώ and gg ο»Ώ are tangent to each other (βˆ‡βƒ—fβˆ₯βˆ‡βƒ—g\vec{\nabla}f \parallel \vec{\nabla}gο»Ώ).

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).