Core Pure Flashcards
What is the inductive statement?
So if true for n = k, then it’s true for n = k + 1
Since true for n = 1, it’s true for all NeZ+
for complex numbers z and w:
what does z*w mean geometrically?
z is enlarged by a s/f of w and rotated through an angle of arg(w) anticlockwise about the origin
for complex numbers z and w:
what does z/w mean geometrically?
z is enlarged by a s/f 1/|w| and rotated through an angle of arg(w) clockwise about the origin
What does a determinant of 1 mean?
the transformation preserves both volume and rotation
only the ______ line is a line of invariant points
mirror
what is the funny e^i0 result?
e^iπ = -1
what is the equation of a line in 3D?
r = a + λn
what is the equation of the tangent plane?
r • n = a • n
n - normal
a - a point on the plane
what does a determinant of -1 mean?
area of the image is the same, but the orientation is reversed
how do you calculate the period of a simple harmonic motion?
period = 2π / ω
for partial fractions problems with denominators including (x² + 4), how do you rewrite them?
the numerator is one order less than the order of the denominator, so it’s Ax + B
for partial fractions problems with denominators including (x + 3)², how do you rewrite them?
rewrite them as A / (x + 3) and B / (x+ 3)²
what’s a standard result for a complex cube root of unity?
1 + ω + ω² = 0
when the roots of the aux equation are DISTINCT REAL ROOTS, what damping does it have?
if the roots are negative, it’s OVERDAMPING
when the roots of the aux equation are a SINGLE REPEATED ROOT, what damping does it have?
if the root is negative, it’s CRITICAL DAMPING
