3.1 KiB
ECE 106: Electricity and Magnetism
MATH 117 review
!!! definition A definite integral is composed of:
- the **upper limit**, $b$,
- the **lower limit**, $a$,
- the **integrand**, $f(x)$, and
- the **differential element**, $dx$.\[\int^b_a f(x)\ dx\]
The original function cannot be recovered from the result of a definite integral unless it is known that \(f(x)\) is a constant.
N-dimensional integrals
Much like how \(dx\) represents an infinitely small line, \(dx\cdot dy\) represents an infinitely small rectangle. This means that the surface area of an object can be expressed as:
\[dS=dx\cdot dy\]
Therefore, the area of a function can be expressed as:
\[S=\int^x_0\int^y_0 dy\ dx\]
where \(y\) is usually equal to \(f(x)\), changing on each iteration.
!!! example The area of a circle can be expressed as \(y=\pm\sqrt{r^2-x^2}\). This can be reduced to \(y=2\sqrt{r^2-x^2}\) because of the symmetry of the equation.
$$
\begin{align*}
A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
&=\int^r_0\sqrt{r^2-x^2}\ dx
\end{align*}
$$!!! 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 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 can be approximated to be a rectangle and is equal to:
\[dS=(dr)(rd\phi)\]
!!! example The area of a circle can be expressed as \(A=\int^{2\pi}_0\int^R_0 r\ dr\ d\phi\).
$$
\begin{align*}
A&=\int^{2\pi}_0\frac{1}{2}R^2\ d\phi \\
&=\pi R^2
\end{align*}
$$So long as the variables are independent of each other, their order does not matter. Otherwise, the dependent variable must be calculated first.
!!! tip There is a shortcut for integrals of cosine and sine squared, so long as \(a=0\) and \(b\) is a multiple of \(\frac\pi 2\):
$$
\int^b_a\cos^2\phi=\frac{b-a}{2} \\
\int^b_a\sin^2\phi=\frac{b-a}{2}
$$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.