# _embedded.expressions[].description_latex
vector form relating force to mass and acceleration

# _embedded.expressions[].latex_rhs
\int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx

# _embedded.expressions[].description_latex
definition of heat capacity at constant volume

# _embedded.expressions[].description_latex
describes a spherical wavefront for an observer in a moving frame of reference

# _embedded.expressions[].latex_lhs
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle

# _embedded.expressions[].latex_rhs
a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle

# _embedded.expressions[].latex_rhs
\frac{1}{V} \frac{nR}{P} \left( \frac{\partial T}{\partial T} \right)_P

# _embedded.expressions[].latex_lhs
\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}

# _embedded.expressions[].name_latex
total resistance for two resistors in parallel

# _embedded.expressions[].description_latex
 the bonding energy in condensed phases is given by the Rydberg energy on the order of several e

# _embedded.expressions[].latex_lhs
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}

# _embedded.expressions[].latex_rhs
\frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)

# _embedded.expressions[].description_latex
define coordinate system such that initial height is at origin

# _embedded.expressions[].latex_rhs
- \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)

# _embedded.expressions[].latex_rhs
\mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)

# _embedded.expressions[].latex_rhs
C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T V \alpha

# _embedded.expressions[].latex_rhs
i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)

# _embedded.expressions[].description_latex
object goes to $\infty$ away from gravitational source

# _embedded.expressions[].latex_rhs
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle

# _embedded.expressions[].latex_lhs
( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle

# _embedded.expressions[].latex_lhs
\langle x^2\rangle -2\langle x \rangle\langle x \rangle+\langle x \rangle^2

# _embedded.expressions[].latex_rhs
KE_{\rm escape} + PE_{\rm Earth\ surface}

# _embedded.expressions[].latex_rhs
\frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)

# _embedded.expressions[].latex_rhs
\left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right)

# _embedded.expressions[].latex_lhs
\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}

# _embedded.expressions[].latex_rhs
\frac{2 \pi r_{\rm Earth\ orbit}}{t_{\rm Earth\ orbit}}

# _embedded.expressions[].description_latex
acceleration is the change in speed over a duration

# _embedded.expressions[].description_latex
acceleration is the average change in speed over a duration

# _embedded.expressions[].latex_rhs
\frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p

# _embedded.expressions[].latex_rhs
-G \frac{m_{\rm Earth} m}{r_{\rm Earth}}

# _embedded.expressions[].latex_lhs
a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle

# _embedded.expressions[].latex_lhs
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}

# _embedded.expressions[].latex_rhs
\psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)

# _embedded.expressions[].latex_rhs
\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar}  \right) \right)

# _embedded.expressions[].latex_rhs
\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)

# _embedded.expressions[].latex_rhs
G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}

# _embedded.expressions[].latex_lhs
i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)

# _embedded.expressions[].latex_lhs
\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)

# _embedded.expressions[].latex_rhs
\frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}

# _embedded.expressions[].latex_rhs
\frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)

# _embedded.expressions[].latex_lhs
\langle x^2\rangle -2\langle x^2 \rangle+\langle x \rangle^2

# _embedded.expressions[].latex_rhs
i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)

# _embedded.expressions[].name_latex
mechanical energy is the sum of the potential plus kinetic energies

# _embedded.expressions[].latex_rhs
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t

# _embedded.expressions[].name_latex
magnitude of vector representation of Newton's second law

# _embedded.expressions[].latex_rhs
G \frac{m_{\rm Earth} m_{\rm satellite}}{r^2}

# _embedded.expressions[].description_latex
definition of internal pressure at constant temperature

# _embedded.expressions[].description_latex
2022-03-25 BHP: Conversion between Latex and Sympy is incomplete

# _embedded.expressions[].latex_rhs
C_V \left(\frac{\partial T}{\partial T}\right)_p + \pi_T \left( \frac{\partial V}{\partial T} \right)_p

# _embedded.expressions[].latex_rhs
G m_{\rm Earth} m \left( \frac{1}{x^2} \bigg\rvert_{ r_{\rm Earth} }^{\infty} \right)

# _embedded.expressions[].latex_rhs
\left( \frac{\partial V}{\partial T} \right)_p

# _embedded.expressions[].description_latex
frame of reference is moving only along x direction

# _embedded.expressions[].description_latex
the potential energy at the surface of the Earth is equal to the work needed to get it from the center of the Earth to the surface

# _embedded.expressions[].latex_lhs
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_lhs
\epsilon^{i,j,k} \hat{x}_i \nabla_j ( \vec{ \nabla} \times \vec{E} )_k

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_lhs
\hat{x}_m \nabla_n \nabla^m E^n - \hat{x}_n \nabla_m \nabla^m E^n

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_lhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_rhs
\vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})

# _embedded.expressions[].latex_rhs
\frac{d}{dt} \left(v_x \hat{x} + v_y \hat{y} \right)

# _embedded.expressions[].latex_rhs
G \frac{m_{\rm Earth} m }{ r_{\rm Earth}}

# _embedded.expressions[].latex_condition
{\rm the\ function\ is\ infinitely\ differentiable\ at}\ x = a.

# _embedded.expressions[].description_latex
not a physically valid result in this context

# _embedded.expressions[].description_latex
intensity of two waves traveling opposite directions on same path

# _embedded.expressions[].latex_lhs
\langle x^2 - 2 x \langle x \rangle + \langle x \rangle^2 \rangle

# _embedded.expressions[].description_latex
frame of reference is moving only along x direction

# _embedded.expressions[].latex_rhs
-k^2 \left(a \sin(kx) + b \cos(kx) \right)

# _embedded.expressions[].description_latex
decompose acceleration into two components

# _embedded.expressions[].description_latex
a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.

# _embedded.expressions[].description_latex
first term was multiplied by \gamma/\gamma

# _embedded.expressions[].latex_lhs
\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t

# _embedded.expressions[].latex_rhs
\int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx

# _embedded.expressions[].description_latex
based on figure 34-27 on page 824 in \cite{2001_HRW}

# _embedded.expressions[].latex_lhs
\theta_{\rm Brewster} + \theta_{\rm refracted}

# _embedded.expressions[].latex_rhs
|Z| |Z| \exp( -i \theta ) \exp( i \theta )

# _embedded.expressions[].latex_lhs
\vec{ \nabla} \times \vec{ \nabla} \times \vec{E}

# _embedded.expressions[].description_latex
proportionality coefficient fvaries in the range 1-4 for a majority of elemental solids

# _embedded.expressions[].description_latex
2022-03-25 BHP: Conversion between Latex and Sympy is incomplete

# _embedded.expressions[].description_latex
a Maclaurin series is a special case of a Taylor series where the expansion point is always x=0.

# _embedded.expressions[].latex_rhs
\frac{-1}{V} \left( \frac{ \partial V}{\partial P} \right)_T

# _embedded.expressions[].latex_rhs
i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)