Summary of 2D Cartesian Coordinates $(x,y)$

Consider the position vector

$${\bf r} = x{\bf i} + y{\bf j}.$$

Assume that both $x$ and $y$ are increased by small amounts $dx$ and $dy$ respectively. Then the point that was orginally at ${\bf r}$ now moves to the new position given by

$${\bf r} + {\bf dr},$$

where

$${\bf dr} = dx{\bf i} + dy{\bf j}.$$ (1.28)

The coefficients of $dx$ and $dy$ are both unity but these factors are called the scale factors. In other coordinate systems the scale factors will be different from unity and they are important in the general forms of div, grad and curl. The infinitesimal area is

$$dS = dx dy$$ (1.29)

and div, grad and curl take the simple form

$$\nabla \cdot {\bf A} = {\partial A_{x}\over \partial x} + {\partial A_{y}\over \partial y}$$

$$\nabla f = {\partial f \over \partial x}{\bf i} + {\partial f \over \partial y}{\bf j}$$

$$\nabla \times {\bf A} = \left ({\partial A_{y}\over \partial x} - {\partial A_{x}\over \partial y}\right ){\bf k}$$