Equations

Symbols

$V_p$

phase voltage

$V_l$

line voltage

$I_l$

line current

$\cos\theta$

power factor

Symbols

$T_0$

the starting time of a cycle

$T$

the periodic time

Symbols

$\epsilon$

permittivity of the material in-between

$d$

distance between the plates

$A$

area of a plate

Symbols

$R$

Resistance

$X$

Reactance

Symbols

$G$

Conductance

$B$

Susceptance

Symbols

$v$

voltage difference across the inductor

$i$

current through the inductor

Symbols

$l$

length

$A$

cross-sectional area

$\rho$

resistivity

Symbols

$Q$

Flow rate

$A$

Cross-sectional area

$v$

velocity

Symbols

$z$

datum head, the height to the point from a reference level

$\frac{P}{\rho g}$

Pressure head

$\frac{v^2}{2g}$

velocity head, the kinetic energy per unit weight

Symbols

$F_S$

Force exerted on the fluid within the control volume

$\dot{M_o}$

Rate of change of momentum of the inflow fluid

$\dot{M_i}$

Rate of change of momentum of the outflow fluid

Symbols

$\dot{M}$

rate of change of momentum

$\rho$

density

$Q$

flow rate

$v$

velocity

Symbols

$\gamma$

specific weight of the fluid

$Q$

flow rate

$H$

total head

Symbols

$K_L$

local loss coefficient

$v$

velocity

$g$

gravitational acceleration

Symbols

$\lambda$

friction factor

$L$

length of the pipe

$D$

diameter of the pipe

$v$

velocity

$g$

gravitational acceleration

Symbols

$H$

provided head

$Q$

flow rate

Symbols

$v_1$

velocity of the jet of fluid

$u$

velocity of the bucket

$k$

loss coefficient (a little less than 1)

$\beta$

angle of deflection of fluid inside the bucket

Symbols

$u$

the upthrust

$v$

the submerged volume

$\rho$

density of the fluid

Symbols

$\omega$

angular velocity

Symbols

$A$

total submerged area

$y_p$

distance to centre of pressure measured along the submerged surface from the free surface

$y_c$

distance to $C$measured along the submerged surface from the free surface

$I_{\text{cc}}$

second moment of submerged area about the centroidal axis parallel to the free surface

Symbols

$h\rho g$

hydrostatic pressure

$h\rho a_z$

due to $a_z$

Symbols

$A$

area of waterline plane

$\overline{x}$

distance to the centroid from axis $OO$

Symbols

$V$

submerged volume

$I$

second moment of area of the waterline plane about the centroidal axis $OO$

Symbols

$\tau$

restoring torque

$W$

Total weight

$I_G$

Moment of inertia of the body about axis of rotation

$M$

Total mass

$k$

Radius of gyration about axis of rotation

Symbols

$D$

domain of $f$

$D'$

codomain of $f$

Symbols

$x = p\cos{\theta}$

real part

$y=p\sin{\theta}$

imaginary part

$p=\sqrt{x^{2} + y^{2}}$

modulus

$\theta=\tan^{-1}{(\frac{y}{x})}$

arg angle

Symbols

$D$

a diagonal matrix

$P$

an invertible matrix

Symbols

$y_p$

**particular solution**

$y_c$

**complementary solution**

Symbols

$c_i$

constant coefficients

$y_i$

a linearly-independent solution

Symbols

$P_0, P_1,\dots,P_n,Q$

differentiable functions of $x$ that do not depend on $y$

$y(x)$

the unknown function

$y^{(n)}$

the $n$th derivative of $y$

Symbols

$x$

a variable

$c$

a constant

Symbols

$\underline{n}$

the unit normal to both $l_1,l_2$

$A, B$

points lying on each line

Symbols

$P$

the arbitrary point

$\underline{p}$

the position vector of $P$

$\underline{r}$

the position vector of a point on the line

$\underline{n}$

parallel to the line

Symbols

$\underline{\dot{\theta}}$

angular velocity of $\text{B}$ relative to $\text{A}$

$v_{\text{B}/\text{A}}$

velocity of $\text{B}$ relative to $\text{A}$

$a_{\text{B}/\text{A}}$

acceleration of $\text{B}$ relative to $\text{A}$

$r_{\text{B}/\text{A}}$

constant position vector of $\text{B}$ relative to $\text{A}$

Symbols

$d$

pitch circle diameter

$m$

pitch of the gear

$z$

no. of teeth in the gear

$r$

radius of the gear

Symbols

$F$

resultant applied force

$m$

mass of the particle

$a$

acceleration of the particle

Symbols

$F$

resultant applied force

$m_i$

mass of a particle

$a_i$

acceleration of a particle

$m$

total mass

$a_G$

acceleration of the center of mass

Symbols

$F$

resultant applied force

$m$

mass of the body

$a$

acceleration of the body

$M$

moment of the resultant force

$I$

mass moment of inertia

$\alpha$

angular acceleration of the body

Symbols

$M_G$

moment about center of mass

$I_G$

mass moment of inertia about center of mass

$\alpha$

angular acceleration of the body

Symbols

$\text{P}$

Center of percussion

$\text{G}$

Center of gravity

$\text{O}$

Pivot

Symbols

$F$

degree of freedoms

$N$

number of kinematic elements

$L$

number of lower pairs

$H$

number of higher pairs

Symbols

$a_1x_1$

the radial component of the relative acceleration between A and B

$x_1b_1$

the transverse component of the relative acceleration between A and B

Symbols

$S_\text{max}$

Maximum stress

$D_\text{max}$

Deflection

$W$

Load

$L$

Span length

$E$

Young's modulus

$I$

Second moment of cross-sectional area

Symbols

$\bar{x}$

Centroid's $x$coordinate

$\bar{y}$

Centroid's $y$coordinate

$A$

Total area

Symbols

$m$

Number of members (bars)

$j$

Number of joints

Symbols

$1$

negative

$0$

positive

Symbols

$V_m, V_f$

volume fractions of matrix and fiber

$E_m, E_f$

elastic moduli of matrix and fiber

Symbols

$M$

molar mass

$n$

atoms per unit cell

$v$

volume of the unit cell

$L$

avagadro's number

Symbols

$\epsilon$

permittivity of the material

$\epsilon_0$

permittivity of free space ($8.85 \times 10^{-12}$F/m)

Symbols

$E_f$

Fermi level

$E_i$

Energy of the $i$th energy state

$K_B$

Boltzmann constant

$T$

Absolute temperature

$\exp()$

exponent function

Symbols

$\rho$

resistivity at temperature $T$

$\rho_0$

resistivity at temperature $T_0$

$\alpha$

temperature coefficient of resistivity

Symbols

$n$

carrier concentration

$q$

charge on a single charge carrier

$\overline{\nu}$

drift velocity

$\sigma$

conductivity

$E$

applied electric field

Symbols

$Q$

charge stored

$d$

charge sensitivity (a material property)

$F$

applied force

Symbols

$V$

applied voltage

$d$

charge sensitivity

$\epsilon$

permittivity

$\sigma$

applied stress

$t$

thickness

Symbols

$\sigma$

conductivity

$n$

carrier concentration

$e$

electron's charge

$\mu_e$

carrier mobility of electrons

$\mu_h$

carrier mobility of holes

Symbols

$P$

fracture load

$L$

support span length

$b$

width of specimen

$d$

depth of specimen

Symbols

$F$

applied load (kgf)

$d$

average diagonal length of indentation (mm)

Symbols

$F$

applied load (in $\text{kN}$)

$D$

diameter of the indenter (in $\text{mm}$)

$d$

diameter of the indentation (in $\text{mm}$)

$h = (D-\sqrt{D^2 - d^2})$

depth of the indentation (in $\text{mm}$)

Symbols

$G$

shear modulus

$K$

bulk modulus

$E$

Young's modulus

Symbols

$C$

Capacitance of the circuit

$V$

Supply voltage

$f$

Charging frequency of the capacitor

Symbols

$V$

Supply voltage

$I_{\text{leakage}}$

Leakage current

Symbols

$x_0, x_1, \ldots, x_n$

the $n+1$distinct data points (nodes)

Symbols

$D$

a diagonal matrix whose entries are of $A$

$-L$

the strictly lower-triangular part of $A$

$-U$

the strictly upper-triangular part of $A$

Symbols

$D$

a diagonal matrix whose entries are of $A$

$-L$

the strictly lower-triangular part of $A$

$-U$

the strictly upper-triangular part of $A$

Symbols

$k$

the degrees of freedom

$\Gamma$

the gamma function

Symbols

$x_i$

each possible value of $X$

$P(X = x_i)$

the probability of observing that value

Symbols

$p$

the probability of success

Symbols

$n$

the number of trials

$\theta$

the probability of success

Symbols

$\lambda$

average number of successes occurred within the fixed time interval

Symbols

$P(A|B)$

the conditional probability of $A$ given $B$

$P(A \cap B)$

the joint probability of both $A$ and $B$occurring

$P(B)$

the probability of event $B$occurring

Symbols

$N$

number of turns

$i$

current in the coil

Symbols

$l$

Length of the path

$\mu$

Permeability

$A$

Cross-sectional area

Symbols

$R$

resistance (ohms)

$C$

capacitance (farads)

$L$

inductance (henries)

Symbols

$t$

time

$s$

$\sigma + j\omega$, where $\sigma, \omega \in \mathbb{R}$

$e^{-st}$

the exponential decay factor that weights the function $f(t)$

Symbols

$R_\text{in}$

Input resistance, very high

$R_\text{out}$

Output resistance, very low

$A$

Gain, ranges from $10^5$ to $10^7$

$V_{d}$

voltage difference between input terminals

$V_{in}$

voltage input, ranges between $5V$ and $18V$

Symbols

$A_v$

Closed-loop voltage gain

Symbols

$Z_m$

the impedance caused by mutual coupling between phases

$Z_s$

the impedance in a single phase

Symbols

$I$

current flow

$R$

resistance of winding

Symbols

$Z_1$

impedance of the primary coil

$Z_2$

impedance of the load in the secondary coil

Symbols

$A$

Voltage gain. Open circuit transfer function. Dimensionless.

$B$

Transfer impedance. Short circuit transfer impedance.

$C$

Transfer admittance. Open circuit transfer admittance.

$D$

Current gain. Short circuit current ratio. Dimensionless.

Symbols

$\nu$

degrees of freedom

$s^2$

sample variance

$\sigma^2$

population variance

Symbols

$H_0$

$\mu = \mu_0$

$H_1$

$\mu \neq \mu_0$

Symbols

$H_0$

$\mu \leq \mu_0$

$H_1$

$\mu \gt \mu_0$

Symbols

$H_0$

$\mu \geq \mu_0$

$H_1$

$\mu \lt \mu_0$

Symbols

$p$

probability of success in a single trial

Symbols

$O_i$

observed frequency for class $i$

$E_i$

expected frequency for class $i$ under $H_0$

$k$

number of non-empty classes

Symbols

$p$

number of parameters estimated from the data

Symbols

$^kC_x$

Number of ways to choose $x$ successes from $k$successes.

$^{N-k}C_{n-x}$

Number of ways to choose $n-x$ failures from $N-k$failures.

$^NC_n$

Total number of ways to choose $n$ items from $N$items.

Symbols

$\alpha$

intercept (value of $Y$ when $X = 0$)

$\beta$

slope (rate of change of $Y$ with respect to $X$)

$\varepsilon_i$

random error, assumed $\varepsilon_i \sim N(0, \sigma^2)$

Symbols

$y_i$

Actual observed value of the dependent variable for observation $i$

$\hat{y_i}$

Predicted value of $y_i$from the regression line

$\bar{y}$

Mean of all observed $y_i$values (overall average)

Symbols

$H_0$

$\mu = \mu_0$

$H_1$

$\mu \neq \mu_0$

Symbols

$H_0$

$\mu \leq \mu_0$

$H_1$

$\mu \gt \mu_0$

Symbols

$H_0$

$\mu \geq \mu_0$

$H_1$

$\mu \lt \mu_0$

Symbols

$p$

probability of success in a single trial

$r$

number of successes required

Symbols

$t$

the value of the random variable

$\nu$

the degrees of freedom (typically, $\nu = n - 1$ for a sample of size $n$)

$\Gamma$

the gamma function

Symbols

$n'$

the successor node of $n$ after applying action $a$

$h(n)$

the heuristic of a node $n$

$h(n')$

the heuristic of a node $n'$

$c(n, a, n')$

the cost of the action $a$ from node $n$ to node $n'$

Symbols

$b$

branching factor

$d$

depth of the shallowest goal

$m$

maximum depth of the search tree

$k$

beam width

Symbols

$S$

the set of states

$A$

the set of actions

$T: S \times A \times S \rightarrow \mathbb{R}_{\ge 0}$

the transition function

$R: S \times A \rightarrow \mathbb{R}_{\ge 0}$

the reward function

Symbols

$s$

current state

$a$

action taken

$s'$

next state

Symbols

$s$

current state

$a$

action taken

$s'$

next state

Symbols

$t$

the current time step

$R_{t+k+1}$

the reward received at time step $t+k+1$

Symbols

$V^*(s)$

the optimal value of $s$

$Q^*(s,a)$

the optimal action-value of $a$ at $s$, the maximum reward for taking an action $a$ in state $s$

Symbols

$P(i)$

probability of outcome $i$

$V_i$

value resulting from that outcome