diff --git a/docs/1b/ece106.md b/docs/1b/ece106.md index 6d4dede..119d24d 100644 --- a/docs/1b/ece106.md +++ b/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**. +