Precalc
Binomial expansion
\begin{align*}
(x+y)^n = (x+y)^{-m} = & x^{-m} \left(1+\frac{y}{x}\right)^{-m} \\
= & x^{-m} \sum_{k=0}^{\infty} {-m \choose k} \left(\frac{y}{x}\right)^k
\end{align*}
where m is positive and x > y and:
\[ {-m \choose k} = (-1)^k {m+k-1 \choose k} \]
.
Whatever’s n:
\[ \binom{n}{k+1} = \frac{n-k}{k+1} \binom{n}{k} \]
Complex & hyperbolic
Identities
\[ e^{i\theta} = \cos{\theta} + i\sin{\theta} \]
\[ z_1z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2+y_1x_2) \]
\[ \abs{z_1z_2} = \abs{z_1}\abs{z_2} \]
\[ \arg(x+iy) = \tan^{-1}{y\over x} \]
\[ \arg(z_1z_2) = \arg{z_1} + \arg{z_2} \]
\[ \frac{z_1}{z_2} = \frac{x_1x_2+y_1y_2}{x_2^2+y_2^2} + i\frac{x_2y_1-x_1y_2}{x_2^2+y_2^2} \]
\[ e^z = \exp z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} \]
De moivre’s
\[ (\cos\theta + i\sin\theta)^n = \cos{n\theta} + i\sin{n\theta} \]
given \(z = e^{i\theta}\)
\[ z^n + \frac{1}{z^n} = 2\cos{n\theta} \]
\[ z^n - \frac{1}{z^n} = 2i\sin{n\theta} \]
Application
\[ \cos^3\theta = \frac{1}{8} \left(z+\frac{1}{z}\right)^3 \]
Roots of unity
Notice
\[ z^n = e^{2ik\pi} = 1 \]
where k is any integer
So the solutions are:
\[ z = e^{2ik\pi/n} \]
Integration
\[ \int e^{ax} cos(bx) \dd x \]
Hyperbolic identities
\[ \cosh x = \frac12(e^x + e^{-x}) \]
\[ \sinh x = \frac12(e^x - e^{-x}) \]
Circular Trig identities hold for hyperbolic functions but replace sinh^2 by -sinh^2 and tanh^2 by -tanh^2
Series
Arithmetic series
\[ \sum_{n=0}^{N-1} (a + nd) \]
Diverges
\[ S_N = \frac{N}{2}[2a + (N-1)d] \]
Geometric series
\[ \sum_{n=0}^{N-1} ar^n \]
\[ S_N = \frac{a(1-r^N)}{1-r} \]
\[ S_\infty = \frac{a}{1-r} \]
Arithmetico-geometric series
\[ \sum_{n=0}^{N-1} (a+nd)r^n \]
\[ S_N = \frac{a-[a+(N-1)d]r^N}{1-r} + \frac{rd(1-r^{N-1})}{(1-r)^2} \]
\[ S = \frac{a}{1-r} + \frac{rd}{(1-r)^2} \]
Difference method
Consider the series \(\sum u_n\)
If u_n can be expressed in the form:
\[ u_n = f(n) - f(n-1) \]term
Notice sum of \(f\) telescopes (\(S_n = f(n)-f(0)\)).
If
\[ u_n = f(n) - f(n-m) \]
then \[ S_N = \sum_{k=1}^m f(N-k+1) - \sum_{k=1}^m f(1-k) \]
Convergence
Real and positive (absolute convergence)
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Prelim test (not sufficient)
\[ \lim_{n\to\infty} u_n = 0. \]
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Comparison test
if \(\sum v_n\) is convergent and
\[ u_n \leq v_n \text{ for } n > N. \]
converse is true
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D’Alembert’s ratio test
\[ \rho = \lim_{n\to\infty} \left(\frac{u_{n+1}}{u_n}\right). \]
if \(\rho < 1\) the series is convergent; if \(\rho > 1\) the series is divergent; if \(\rho = 1\), it’s undetermined by this test
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Ratio comparison test
\(v_n\) is convergent and
\[ \frac{u_{n+1}}{u_n} \leq \frac{v_{n+1}}{v_n} \]
for all n greater than a fixed value N
converse is true
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Quotient test
\[ \rho = \lim_{n\to\infty}\left(\frac{u_n}{v_n}\right) \].
if \(\rho \neq 0\) and finite: both converge or both diverge; if \(\rho = 0\) and \(\sum v_n\) converges: then \(\sum u_n\) converges; if \(\rho = \infty\) and \(\sum v_n\) diverges: then \(\sum u_n\) diverges;
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Integral test
if the limit of the integral
\[ \lim_{N\to\infty} \int^N U(x) \dd{x} \]
exists, the series \(\sum u_n\) is convergent.
the lower limit is not stated to avoid silly cases like singularities
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Cauchy’s root test
\[ \rho = \lim_{n\to\infty} (u_n)^{1/n}. \]
if \(\rho < 1\) the series is convergent; if \(\rho > 1\) the series is divergent; if \(\rho = 1\), it’s undetermined by this test;
good for when the nth term contains an nth power like \(\sum_{n=1}^{\infty} \left(\frac{1}{n}\right)^n\)
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Grouping
trivial
Otherwise (merely convergent)
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Alternating series test
\[ \sum_{n=1}^\infty (-1)^{n+1} u_n \]
where \(u_n \geq 0\).
It converges when:
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\(u_n \to 0\) as \(n \to \infty\)
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\(u_n < u_{n-1}\) for all \(n>N\) for some finite \(N\).
otherwise it oscillates
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Operations
If the infinite series \(\sum u_n\) and \(\sum v_n\) are both absolutely convergent then the series \(\sum w_n\), where:
\[ w_n = u_1v_n + u_2v_{n-1} + \ldots + u_nv_1 \]
is also absolutely convergent.
\(\sum w_n\) is the Cauchy product and it converges to ST is u converges to S and v converges to T;
Power series
Converges if:
\[ \rho = \abs{x} \lim_{n\to\infty} \abs{\frac{a_{n+1}}{a_n}} < 1 \]
Taylor series
\[ f(a+h) = \frac{h^{n-1}}{(n-1)!}f^{(n-1)}(a) + R_n(h) \]
\[ R_n(h) = \frac{h^n}{n!}f^{(n)}(\zeta) \]
for some \(\zeta \in [a,a+h]\)
Partial derivatives
Total differential
\[ \dd f = \pdv{f}{x}\dd x + \pdv{f}{y}\dd y \]
If \(x_i = x_i(x_1), i = 2,3,\ldots,n\), then the total derivative is:
\[ \dv{f}{x_1} = \pdv{f}{x_1} + \left(\pdv{f}{x_2}\right)\dv{x_2}{x_1} + \ldots + \left(\pdv{f}{x_n}\right) \dv{x_n}{x_1} \]
Pretend that the other variables do not change with \(x_1\) when evaluating \(\pdv{f}{x_1}\)
Exact differentials
A differential is exact if when given:
\[ \dd f = A(x,y)\dd x + B(x,y) \dd y \],
\[ \pdv{A}{y} = \pdv{B}{x}, \]
or
\[ \pdv{g_i}{x_j} = \pdv{g_j}{x_i} \text{ for all pairs $i,j$}. \]
and there are \(\frac12 n(n-1)\) relationships to be satisfied
Useful theorems
Reciprocity relation:
\[ \left(\pdv{x}{y}\right)_z = \left(\pdv{y}{x}\right)_z^{-1} \]
given both exist and neither of zero
Cyclic relation:
\[ \left(\pdv{y}{z}\right)_x \left(\pdv{z}{x}\right)_y \left(\pdv{x}{y}\right)_z = -1 \]
Chain rule
\[ \dv{f}{u} = \pdv{f}{x}\dv{x}{u} + \pdv{f}{y}\dv{y}{u} \]
Good for parametric equations without having to express x(u) and y(u) in terms of u.
Can be extended to problems requiring change of variable by replacing the total derivative with partial ones
Taylor’s theorem
\begin{align*}
f(x,y) = f(x_0,y_0) & + \pdv{f}{x}\Delta x + \pdv{f}{y} \Delta y \\
& + \frac{1}{2!}\left[\pdv[2]{f}{x}(\Delta x)^2 + 2\pdv[2]{f}{x}{y}\Delta x \Delta y + \pdv[2]{f}{y}(\Delta y)^2\right] + \ldots \\
\end{align*}
\[ = \sum_{n=0}^{\infty}\frac{1}{n!}\left[\left(\Delta x \pdv{x} + \Delta y \pdv{y}\right)^n f(x,y)\right]_{x_0,y_0} \]
or more generally:
\[ f(\textbf x) = f(\textbf{x}_0) + \sum_i \pdv{f}{x_i}\Delta x_i + \frac{1}{2!}\sum_i\sum_j\pdv[2]{f}{x_i}{x_j}\Delta x_i \Delta x_j + \ldots \] \[ = \sum_{n=0}^\infty \frac{1}{n!}\left[\left(\Delta \textbf{x} \cdot \nabla\right)^n f(\textbf x)\right]_{\textbf{x}=\textbf{x}_0} \]
Stationary values
all stationary points have \(f_x = f_y = 0\) minima if \(f_{xx}\) and \(f_{yy}\) are positive and \(f_{xy}^2 < f_{xx}f_{yy}\) maxima if \(f_{xx}\) and \(f_{yy}\) are negative and \(f_{xy}^2 < f_{xx}f_{yy}\) saddle points if \(f_{xx}\) and \(f_{yy}\) have opposite signs or \(f_{xy}^2 > f_{xx}{yy}\)
Stationary values under constraints (lagrange multipliers)
Abridged problem statement: Find the maximum value of the differentiable function f(x,y) subject to the constraint g(x,y) = c, where c is a constant.
We must choose a \(\lambda\) (Lagrange undetermined multiplier) such that:
\[ \pdv{f}{x} + \lambda\pdv{g}{x} = 0 \]
\[ \pdv{f}{y} + \lambda\pdv{g}{y}=0 \]
this can be extended to multiple constraints given the no. of contraints is smaller than the number of variables
Envelope
A function where
\[ f(x,y,\alpha_1) = 0 \]
and
\[ \pdv{f(x,y,\alpha_1)}{\alpha}=0 \]
Differentiation of integrals
\[ \dv{x}\left[\int_u^v f(x,t) \dd{t} \right] = \int_u^v \pdv{f(x,t)}{x}\dd{t} \]
given v and u are constants
\[ \dv{x}\left[\int_{u(x)}^{v(x)} f(x,t) \dd{t}\right] = f(x,v(x))\dv{v}{x} - f(x,u(x))\dv{u}{x} + \int_{u(x)}^{v(x)}\pdv{f(x,t)}{x}\dd{t} \]
Multiple Integrals
Center of mass
\[ \bar{x} \int \dd M = \int x \dd M \]
Pappus theorems
Volume generated by rotating the plane area about the x-axis
\[ V = \int 2\pi y \dd{A} = 2\pi\bar{y}A \]
Area generated by rotating the plane curve about the x-axis (surface of revolution): \[ S = \int 2\pi y \dd{s} = 2\pi\bar{y}L \]
Change of variables (Jacobian)
The Jacobian is defined by
\[
J = \frac{\partial(x,y)}{\partial(u,v)} =\begin{vmatrix}
\pdv{x}{u} & \pdv{y}{u} \\
&\\
\pdv{x}{v} & \pdv{y}{v} \\
\end{vmatrix}
\]
Thus:
\[ \dd{x} \dd{y} = \left|J\right| \dd{u} \dd{v} \]
Note that:
\[ \dd{x}\dd{y} = r \dd{r} \dd{\phi} \]
Or that: \[ \dd{V} = r^2\sin\theta \dd{r}\dd{\theta}\dd{\phi} \]
Also:
\[ J_{xy}J_{yx} = 1 \]
or:
\[ \frac{\partial(x_1,\ldots,x_n)}{\partial(y_1,\ldots,y_n)} = \left[\frac{\partial(x_1,\ldots,x_n)}{\partial(y_1,\ldots,y_n)}\right]^{-1} \]
Vector algebra
\[ (a\times b)\cdot(c\times d) \equiv (a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c) \]
\[ a\times(b \times c) = (a \cdot c)b - (a \cdot b)c \]
\[ (a\times b)\times c = (a \cdot c)b - (b \cdot c)a \]
Ratio theorem
\[ \textbf{OP} = \frac{\mu}{\lambda+\mu}\textbf{a} + \frac{\lambda}{\lambda+\mu}\textbf{b} \]
Geometry
Line
\[ {\bf r} = {\bf a} + \lambda{\bf b} \]
\[ ({\bf r - a}) \times {\bf b} = 0 \]
Plane
\[ ({\bf r - a}) \cdot {\bf \hat n} = 0 \]
\[ {\bf r} \cdot {\bf \hat n} = d \]
\[ {\bf r} = \alpha {\bf a} + \beta{\bf b} +\gamma {\bf n} \]
where a + b + c = 1
lx + my + nz = d
Sphere
| r-c | ^2 = a^2 |
Normal modes
The potential must not depend on the time derivatives of these of these coordinates. The potential must have a local minimum at the equillibrium point. The potential at this point is zero.
T can be written as the quadratic form.
\[ T = \sum_i \sum_j a_{ij} \dot{q}_i \dot{q}_j = \dot{q}^TA\dot{q} \]
and likewise for V with q instead of qdot.
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A and B can be made symmetric.
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A and B are real and positive definite.
Process
We may use the above equation to write the equation E = T + V.
In the above case:
\begin{aligned}
T & \approx \frac12 M l^2 (\dot{\theta}_1^2 + \frac{1}{4} \dot{\theta}_2^2 + \dot{\theta}_1 \dot{\theta}_2) + \frac{1}{24}M l^2 \dot{\theta}_2^2 \\
& = \frac{1}{12} M l^2 \dot{q}^T \begin{pmatrix} 6 & 3 \\ 3 & 2 \\end{pmatrix}\dot{q}
\end{aligned}
\begin{aligned}
V & \approx \frac{1}{4} M l g (2\dot{\theta}_1^2 + \dot{\theta}_2^2 \\
& = \frac{1}{12} M l g q^T \begin{pmatrix} 6 & 0 \\ 0 & 3 \\end{pmatrix}q
\end{aligned}
We may then apply conservation of energy \(\dv{t} (T+V) = 0\).
This simplifies* to \(A\ddot{q} + Bq = 0\). Solutions of this form satisfy \(q = x\cos\omega t\)
There will be N values of \(\omega\) and these values are (normal || eigen)frequencies. This has non-trivial solutions only if:
\[ \vert B - \omega^2 A \vert = 0 \]
Note that the eigenvectors are mutually orthogonal (except when degeneracy) as A and B are real symmetric matrices.
It is possible to get multiple normal modes by exploiting symmetries and then using the fact that the eigenvectors are orthogonal.
Rayleigh-Ritz method
Vector calculus
Differentiation of a vector function
If magnitude \(|\textbf{a}|\) is constant wrt u
\[ \textbf{a} \cdot \dv{\textbf{a}}{u} = 0 \]
A function can be represented as:
\[ \textbf{r}(u) = x(u)\textbf{i} + y(u)\textbf{j} + z(u)\textbf{k} \]
\[ \textbf{r}(u) = u\textbf{i} + f(u)\textbf{j} + g(u)\textbf{k} \]
Arc length
the arc length between two points on the curve \({\bf r }(u)\) is given by:
\[ s = \int_{u_i}^{u_2} \sqrt{\dv{\bf r}{u} \cdot \dv{\bf r}{u}} \dd{u} \]
tangent, binormal and normal (???)
If the parameter u of r(u) is the arc length along the curve then dr/ds is the unit tangent vector \(\hat{t}\)
The curvature (\(\kappa\)) or the radius of curvature (\(\rho\))is defined by:
\[ \kappa = \frac{1}{\rho} =\left|\dv{\hat{t}}{s}\right| = \left|\dv[2]{\hat{r}}{s}\right| \]
The unit vector perpendicular to \(\hat{t}\) is denoted by \(\hat{n}\), and is the principal normal.
Thus:
\[ \dv{\hat{\bf t}}{s} = \kappa\hat{\bf n} \]
The unit vector \(\bf \hat{b} = \hat{t} \times \hat{n}\) is called the binormal to C, and t,n,b thus form a right handed
rectangular coordinate system.
The torsion (\(\tau\)) is given by \(\dv{\hat b}{s}\) and is related to the normal by \(\dv{b}{s} =-\tau \hat n\) or :
\[ \tau = -\hat{\bf n} \cdot \dv{\hat{\bf b}}{s} \]
There is also the radius of torsion, \(\sigma = 1/\tau\)
\[ \dv{\hat{\bf n}}{s} = \tau \hat{\bf b} - \kappa \hat{\bf t} \]
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Frenet-Serret formulae
\[ \dv{\hat{\bf t}}{s} = \kappa \hat{\bf n} \]
\[ \dv{\hat{\bf n}}{s} = \tau \hat{\bf b} - \kappa \hat{\bf t} \]
\[ \dv{\hat{\bf b}}{s} =-\tau \hat{\bf n} \]
Surfaces
a surface can be given by:
\[ {\bf r}(u,v) = u{\bf i} + v{\bf j} + f(u,v){\bf k} \]
Coordinate curves are curves with u= constant or v = constant. Notice that the tangent vector of u=constant is in direction dr/du
A normal to the surface at P is given by:
\[ {\bf n} = \pdv{\bf r}{u} \times \pdv{\bf r}{v} \]
or
\[ \hat{\bf n}(x,y,z) = \frac{\pdv{r}{u}\times\pdv{r}{v}}{|\pdv{r}{u}\times\pdv{r}{v}|} = \frac{-\bf{i}\pdv{f}{x}-\bf{j}\pdv{f}{y}+\bf{k}}{\sqrt{1 + \left(\pdv{f}{x}\right)^2 + \left(\pdv{f}{y}\right)^2}} \] The element of area at P:
\[ \dd{S} = |{\bf n}| \dd{u} \dd{v} \]
Thus:
\[ A = \iint_R |{\bf n}| \dd{u}\dd{v} \]
Surface integrals of normal component of function F
\[ \iint_S \bf{F\cdot \hat{n}} \dd{S} = \lim_{N\to\infty, \Delta S \to 0} \sum_{l=1}^N \bf{F}(x_l,y_l,z_l) \bf{\cdot \hat{n}_l} \Delta S_l \]
The surface area is given by:
\[ \iint_S \dd{S} \]
Integrating over S -> Integrating over projection of S
We find:
\[ \hat{\bf n} \cdot \bf{k}=\frac{1}{\sqrt{1+\left(\partial{f}/\partial{x}\right)^2 + \left(\partial{f}/\partial{y}\right)^2}} \]
We can then write:
\[ \iint_S G(x,y,z) \dd{S} = \iint_R G[x,y,f(x,y)] \cdot \sqrt{1 + \left(\pdv{f}{x}\right)^2 + \left(\pdv{f}{y}\right)^2} \dd{x}\dd{y} \]
which allows to integrate G over the projection on the x-y plane. This can be generalised to integration on the y-z plane and x-z plane.
Flux of F
To integrate integrals of the form \(\bf F \cdot \hat{n}\)
we can simplify the earlier result into:
\begin{align*} \iint_S \bf{F\cdot \hat{n}} \dd{S} & = \iint_R -F_x[x,y,f(x,y)]\pdv{f}{x} \\ - & F_y[x,y,f(x,y)]\pdv{f}{y}+ F_z[x,y,f(x,y)] \dd{x} \dd{y} \end{align*}
The above integral is often called the flux of \(\bf F\). The rate of flow through a surface S can be given by subsitiuting F for \(\rho(x,y,z){\bf v} (x,y,z)\)
Vector operators
Nabla
Gives a vector field
Cartesian:
\[ \nabla \equiv {\bf \hat i}\pdv{x} + {\bf \hat j}\pdv{y} + {\bf \hat k}\pdv{z} \]
Divergence
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Definitions
\[ \div {\bf F} = \lim_{\Delta V \to 0} \frac{1}{\Delta V} \iint_S {\bf F \cdot \hat{n}} \dd{S} \]
Where \(\Delta V\) is taken about (x,y,z)
This allows us to express gauss' law as \(\div E = \rho/\epsilon_0\)
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Cartesian
\[ \div F = \pdv{F_x}{x} + \pdv{F_y}{y}+ \pdv{F_z}{z} \]
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Cylindrical
\[ \div F = \frac{1}{r}\pdv{}{r}\left(rF_r\right) + \frac{1}{r}\pdv{F_\theta}{\theta}+ \pdv{F_z}{z} \]
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Spherical
\[ \div F = \frac{1}{r^2}\pdv{}{r}\left(r^2F_r\right) + \frac{1}{r \sin\phi}\pdv{}{\phi} \left( \sin\phi F_\phi \right) + \frac{1}{r \sin\phi}\pdv{F_\theta}{\theta} \]
-
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Divergence theorem
\[ \iint_S {\bf F \cdot \hat{n}} \dd{S} = \iiint_V {\div {\bf F}} \dd{V} \]
Grad of a scalar field
Rate of change of \(\phi\) (a scalar field) wrt distance s in direction \(\bf a\) (i.e. the directional derivative)
\[ \dv{\phi}{s} = \nabla \phi \cdot {\bf \hat a} \]
To find the rate of change of a vector field in a particular direction: \[ \hat{\bf a} \cdot \nabla = a_x \pdv{x} + a_y \pdv{y} + a_z\pdv{z} \]
e.g. \(\dd{\bf E} = (\dd{\bf r} \cdot \nabla){\bf E}\)
A second interesting property is that: \(\nabla\phi\) is a vector normal to the surface \(\phi(x,y,z) = c\) at every point