Properties:
- Path independent
- They have an associate (scalar) potential (energy)
- The work done equals the change in potential energy
- Change in potential energy is matches by change in kinetic energy
$$ \vec \nabla \times \vec F = 0 $$
$$ \vec{F} = -\vec{\nabla} E_p $$
$$ W = \int_C \vec{F} \cdot \boldsymbol{dl} = \int_{A}^{B} -\boldsymbol{\nabla} V \cdot \text d\vec{l} = -\int_{A}^{B} \text dV = V_A - V_B $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/724d410b-8e31-408d-a9ac-ef874d495ce0/Central_forces.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/724d410b-8e31-408d-a9ac-ef874d495ce0/Central_forces.png" width="40px" /> Central forces: forces which depend only on $r$ and are in the $\hat r$ direction
$$ \vec F=f(r) \,\hat r $$
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$$ E_p=-\int F_x\,\text dx=-\int F_y\,\text dy=-\int F_z\,\text dz $$
Dirichlet Conditions for Fourier series
- $P(t)$ must be ‘bounded’ i.e. have a finite value over the whole period
- $P(t)$ must only have a finite number of discontinuities and min/maxima over the period
- $P(t)$ must be integratable over the period of the function
$$ f=\frac{1}{T} \qquad ; \qquad \omega=2\pi f=\frac{2\pi}{T} $$
🗒️ A function is period with period $T$ if:
$$ \int_0^T P(t)\,\text dt = \int_{t_0}^{T+t_0} P(t)\,\text dt $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8d9264bd-8305-4011-9db1-3a2dd1c6198a/orthogonal_function.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8d9264bd-8305-4011-9db1-3a2dd1c6198a/orthogonal_function.png" width="40px" /> Orthogonal function: $f(t)$ and $g(t)$ are orthogonal if
$$ \int_0^T f^*(t) g(t)\,\text dt = 0 $$
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