65 lines
1.5 KiB
Plaintext
65 lines
1.5 KiB
Plaintext
#import "@preview/cetz:0.4.2": canvas, draw
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#import "@preview/cetz-plot:0.1.3": plot
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#import "../template/lib.typ": *
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#set page(paper: "a4")
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#show: notes.with(
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title: [EE1P1],
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subtitle: [Electricity and Magnetism],
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author: "Folkert Kevelam"
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)
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= Lecture 1 - Mathematical Instruments
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== Vectors
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#definition[
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definition for a vector with a given source point by $arrow(r)_1$
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$
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arrow(r)_(1,2) &= (x_2-x_1)hat(x) + (y_2-y_1)hat(y) + (z_2-z_1)hat(z) \
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|arrow(r)_(1,2)| &= sqrt((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2) \
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hat(r)_(1,2) &= arrow(r)_(1,2) / (|arrow(r)_(1,2)|)
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$
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Most of the time, vectors are defined from a source point to a observation
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point.
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$
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arrow(R) = arrow(r) - attach(limits(r), t: arrow prime)
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$
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]
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#definition[
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Scalar product: $RR^3 dot RR^3 arrow RR$, $arrow(a)dot arrow(b) = arrow(b) dot arrow(a)$
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Cross product: $RR^3 times RR^3 arrow RR^3$, $arrow(a) times arrow(b) = -arrow(b) times arrow(a)$
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$arrow(a) times arrow(b) = ||$
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$|arrow(a) times arrow(b)| = |arrow(a)||arrow(b)| sin(alpha)$
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$arrow(a) dot (arrow(b) times arrow(c)) = arrow(c) dot (arrow(a) times arrow(b))$
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$arrow(a) times (arrow(b) times arrow(c))$
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]
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== Integrals
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=== Integrals over $cal(D) subset RR^3$
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+ Compact form: $integral_cal(D) f(x,y,z) d V$
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+ Extended form: $integral.triple_cal(D) f(x,y,z) d V$
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=== Polar coordinates
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+ EE1M1: $r, theta$, EE1P1: $rho, phi$
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=== Cylindrical coordinates
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+ EE1M1: $r, theta, z$, EE1P1: $rho, phi, z$
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=== Spherical coordinates
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+ EE1M1: $r, theta, phi$, EE1P1: $rho, phi, theta$
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