The diagonal of the rectangle on the left below shows (a blown-up picture of) an infinitesimal displacement from the point (\(x\), \(y\)) to the nearby point
(\(x+dx\), \(y+dy\)).

• Label the rectangle with the lengths of the sides.

• Express the sides of the rectangle indicated by arrows as vectors.

  Use the unit vectors \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).

• The diagonal of this rectangle is the vector differential \(d\vec{r}\). Express \(d\vec{r}\) in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\).
• Find the length \(ds=|d\vec{r}|\) of the diagonal.

• Label the rectangle with the lengths of the sides. Careful!
• Express the sides of the rectangle indicated by arrows as vectors.
  Use the natural orthonormal basis defined by the picture, that is, let \(\hat{r}\) be the unit vector which points in the direction of increasing \(\vec{r}\), and let \(\hat{\phi}\) be the unit vector which points in the direction of increasing \(\phi\). Do not attempt to express these vectors in terms of \(\boldsymbol{\hat{x}}\) and \(\boldsymbol{\hat{y}}\)! You do not need to worry about the fact that some sides of the rectangle aren't straight; the rectangle is so small that this error is negligible.
• The diagonal of this rectangle is again the vector differential \(d\boldsymbol{\vec{r}}\). Express \(d\boldsymbol{\vec{r}}\) in terms of \(\hat{r}\) and \(\hat{\phi}\)
• Find the length \(ds=|d\vec{r}|\) of the diagonal.