Multi-variable Calculus


Chain Rule with More Variables

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

$$dw = f_x dx + f_y dy = (f_xx_u+ f_yy_u)du + (f_xx_v+ f_yy_v)dv = \frac{\partial f}{\partial u}du + \frac{\partial f}{\partial v}dv$$

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

Directional Derivatives

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


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

Lagrange Multipliers

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

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

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

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

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).