0. MathematicsSavchenko Formulas, chapter 0 of 14, 11 formulas

$\alpha$

angle in radians, $\alpha \ll 1$

True only for an angle in radians. Small pendulum oscillations, paraxial rays and small displacements from equilibrium all reduce to these replacements. When the answer holds $1 - \cos\alpha$, the first approximation $\cos\alpha \approx 1$ is not enough and the quadratic term is needed.

x

small quantity

$\alpha$

any exponent, whole, fractional or negative

The main estimating tool of the book. A difference of two close quantities, for example $\sqrt{l^2 + d^2} - l \approx d^2/2l$, is found by taking the large one outside the brackets and expanding the rest. The next term is $\alpha(\alpha - 1)x^2/2$, used when the first one cancels.

x

small quantity

This is how an exact formula with a logarithm or an exponent (barometric, isothermal work, capacitor discharge) turns linear when the change is small.

x₀

initial value

$\tau$

time to fall $e$ times

When the rate of decrease is proportional to the quantity itself, separate the variables, $dx/x = -dt/\tau$, and integrate, $\ln(x/x_0) = -t/\tau$. That is how a capacitor discharges, oscillations die out, light is absorbed, and a rope round a post gives $F = F_0e^{\mu\alpha}$.

$f(x)$

function of a quantity that depends on time

The rate of change of a quantity that depends on an angle or a coordinate is its derivative by that variable times the variable's rate. So $x = A\sin\omega t$ gives $v = A\omega\cos\omega t$, and $y = f\tan\gamma$ gives the spot speed $f\dot\gamma/\cos^2\gamma$.

a_x, a_y

mutually perpendicular components

Only perpendicular components add this way, for example the tangential and normal accelerations or the velocities of two perpendicular motions. Vectors at an angle need the law of cosines.

a, b

magnitudes of the vectors

$\alpha$

angle between the vectors

This is the law of cosines for the triangle made of the vectors. In momentum conservation for a break-up or an oblique collision it relates the magnitudes of the momenta without components.

r

radius

The area of a sphere is the derivative of the volume of a ball by its radius, so a thin spherical shell has mass $4\pi r^2\rho\,dr$. In terms of the diameter the area of a circle is $\pi d^2/4$.

$\alpha, \beta$

angles

Needed when adding oscillations, for the range $l = v^2\sin2\alpha/g$ and when turning to rotated axes.

a, b, c

sides of the triangle

$\alpha, \beta, \gamma$

opposite angles

Handy in a velocity triangle when the directions are known but not the magnitudes, for example crossing a river or adding the wind and aircraft velocities.

a, b, c

coefficients

Two roots mean two physical answers or one extra root dropped by its meaning (a negative time, a speed above the initial one). The condition for the roots to exist, $b^2 \geq 4ac$, is often the answer to a question about the least speed or the limiting angle.