ece106: day 1 update

docs/1b/ece106.md

@@ -39,4 +39,37 @@ where $y$ is usually equal to $f(x)$, changing on each iteration.
 !!! warning
     Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.
 
+These rules also apply for a system in three dimensions:
+
+| Vector | Length | Area | Volume | 
+| --- | --- | --- | --- |
+| $x$ | $dx$ | $dx\cdot dy$ | $dx\cdot dy\cdot dz$ |
+| $y$ | $dy$ | $dy\cdot dz$ | |
+| $z$ | $dz$ | $dx\cdot dz$ | |
+
+Although differential elements can be blindly used inside and outside an object (e.g., area), the rules break down as the **boundary** of an object is approached (e.g., perimeter). Applying these rules to determine an object's perimeter will result in the incorrect deduction that $\pi=4$.
+
+Therefore, further approximations can be made by making a length $\dl=\sqrt{(dx)^2+(dy)^2}$ to represent the perimeter.
+
+!!! example
+    This reduces to $dl=\sqrt{\left(\frac{dy}{dx}\right)^2+1}$.
+
+### Polar coordinates
+
+Please see [MATH 115: Linear Algebra#Polar form](/1a/math115/#polar-form) for more information.
+
+In polar form, the difference in each "rectangle" side length is slightly different.
+
+| Vector | Length difference |
+| --- | --- |
+| $\hat r$ | $dr$ |
+| $\hat\phi$ | $rd\phi$ |
+
+Therefore, the change in surface area is equal to:
+
+$$dS=(dr)(rd\phi)$$
+
 ## Cartesian coordinates
+
+The axes in a Cartesian coordinate plane must be orthogonal so that increasing a value in one axis does not affect any other. The axes must also point in directions that follow the **right hand rule**.
+