Ordinary Diff Eqs
for the linear ODE y'' + p(x) y' + q(x) y = r(x)
if r(x) = 0 we say ...
The ODE is homogeneous
u(0,t) = 0 and u(L,t) = 0 are known as the
Dirichlet Boundary Conditions
Name three types of convergence tests
Integral test
Ratio test
Root test
Alternating Series test
Is there only one allowable way to parameterize an integral problem?
No. Parameterization is always a choice. Sometimes there is an obvious or best choice.
the equations u_x = v_y u_y = - v_x
are
The Cauchy Riemann Equations, used to verify whether a complex function is analytic (differentiable)
y'' + a y' + b y = 0 has as it's solution basis...
Two linearly-independent eigen-functions
e^(lambda_1 x)
e^(lambda_2 x) or x e^(lambda_1 x)
Note that they may be re-written to use sin and cos if the lambdas are complex
For the time-evolution of modes in a Heat Equation problem,
the G_n(t) terms are of the form...
Exponential Decay Function
sum of 1/n for n from 0 to infinity is known as ...
the Harmonic Series It does not converge to a finite sum.
For a line integral, the r'(t) expression is
instantaneous tangent to the curve
The difference between arg z and Arg z is
an integer multiple of 2*pi
For y'' + a y' + b y = r(t) where a and b are real (scalar) coefficients
if a^2 < 4 b then the eigenvalues will be
a complex-conjugate pair
oscillate faster (think higher frequency)
sum of x^n / n! for n = 0..infinity is recognized as the
exponential function of x
Name three re-parameterizations commonly seen in vector calculus
polar coordinates
cylindrical coordinates
spherical coordinates
The roots of complex numbers are more easily computed by...
converting the complex number to polar form
|z| ( cos Arg (z) + i sin Arg(z) )
For y'' + a y' + b y = r(t)
we call the expression
1 / (s^2 + a s + b)
the Transfer Function
If the divergence of the gradient of a scalar-valued function f(x,y,z) equals zero, we say that
f is a potential function
solenoidal
Better than a lower-rectangle integration or an upper-rectangle integration is a integration using trapezoids... often better than that is a method called Simpson's method... what makes it better?
It integrates with quadratic segments
when you see a ds in an integral it means
The s parameter is proportional to the length of the curve, it is length-preserving. You'll see s in the integral for Arc Length.
The zero result of a closed path integral of 1 / z^2 around the origin (0,0) tells us
Not all nonanalytic functions produce a residue
Cauchy's Integral Theorem (analyticity) is a sufficient but not necessary condition for the zero integration result
An input oscillation to a 2nd-order system (ODE) will produce
(assume steady state)
an oscillation with the same frequency, but it may be altered by amplitude and incur a phase shift
The D-Alembert solution might look like a travelling wave going in only one direction if
The initial condition includes both an initial displacement and an initial velocity
A discontinuity in the function f(x) leads to a Fourier Series with a little tail near the discontinuity ...
Gibb's Effect
True or False, if you parameterize by x and y, you will never have to compute a Jacobian determinant
False. If the surface is not on the x,y plane, a Jacobian Determinant will still be necessary, but it's value is obtained in the magnitude of the r_u X r_v normal vector when calculated.
finding the value of an analytic function at a given point