76 lines
2.5 KiB
Markdown
76 lines
2.5 KiB
Markdown
# ECE 106: Electricity and Magnetism
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## MATH 117 review
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!!! definition
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A definite integral is composed of:
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- the **upper limit**, $b$,
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- the **lower limit**, $a$,
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- the **integrand**, $f(x)$, and
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- the **differential element**, $dx$.
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$$\int^b_a f(x)\ dx$$
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The original function **cannot be recovered** from the result of a definite integral unless it is known that $f(x)$ is a constant.
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## N-dimensional integrals
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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:
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$$dS=dx\cdot dy$$
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Therefore, the area of a function can be expressed as:
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$$S=\int^x_0\int^y_0 dy\ dx$$
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where $y$ is usually equal to $f(x)$, changing on each iteration.
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!!! example
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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.
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$$
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\begin{align*}
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A&=\int^r_0\int^{\sqrt{r^2-x^2}}_0 dy\ dx \\
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&=\int^r_0\sqrt{r^2-x^2}\ dx
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\end{align*}
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$$
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!!! warning
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Similar to parentheses, the correct integral squiggly must be paired with the correct differential element.
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These rules also apply for a system in three dimensions:
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| Vector | Length | Area | Volume |
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| --- | --- | --- | --- |
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| $x$ | $dx$ | $dx\cdot dy$ | $dx\cdot dy\cdot dz$ |
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| $y$ | $dy$ | $dy\cdot dz$ | |
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| $z$ | $dz$ | $dx\cdot dz$ | |
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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 $\pi=4$.
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Therefore, further approximations can be made by making a length $\dl=\sqrt{(dx)^2+(dy)^2}$ to represent the perimeter.
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!!! example
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This reduces to $dl=\sqrt{\left(\frac{dy}{dx}\right)^2+1}$.
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### Polar coordinates
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Please see [MATH 115: Linear Algebra#Polar form](/1a/math115/#polar-form) for more information.
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In polar form, the difference in each "rectangle" side length is slightly different.
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| Vector | Length difference |
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| --- | --- |
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| $\hat r$ | $dr$ |
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| $\hat\phi$ | $rd\phi$ |
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Therefore, the change in surface area is equal to:
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$$dS=(dr)(rd\phi)$$
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## Cartesian coordinates
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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**.
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