Pure Flashcards
Critical damping
Discriminant of AE is 0
Underdamping
Discriminant is negative
Overdamping
Discriminant is positive
Scaler product
a.b=|a|.|b|cosx
Vector equation of a plane
(r-a).n=0
Cartesian equation of a plane
n1x+n2y+n3z+d=0
Find angle between two planes
Use scalar product on normals
Vector equation of a line in 2d
R=a+¥d
Cartesian equation of line 2d same for 3d
¥= x-a/d=y-a1/d1
Special case for Cartesian equation of a line in 3D
When d=0 write as x=a , ¥=y-a1/d1
Intersection of two lines in 2d
Solve direction simultaneously
How to prove to lines in 2d are skew
No unique solution simultaneously
Enlargement scale factor k about origin
(K 0)
O k
Stretch scale factor m parallel to x axis
(m 0)
0 1
Stretch scale factor n parallel to y axis
(1 0)
0 n
Rotation 90 degrees clockwise about origin and any rotation about origin through x degrees
(0 1)
(-1 0)
(Cosx -sinx)
(Sinx cosx)
Reflection in x and y axis
X axis (1 0)
(0 -1)
Y axis (-1 0)
(0 1)
Reflection y=x
(0 1)
1 0
Equation of a shear
Shear x axis fixed (1 k)
(0 1)
Y axis fixed (1 0)
(k 1)
How do you rotate 3D shapes
Check book if unsure
What is an invariant point
A point that doesn’t change when multiplied by a matrix
How to find a line of invariant points
Set up a matrix equation will lead to two equations like ax + by = 0, if unique solution then line of invariant points, if not origin only point of invariance
What is an invariant line
Any point of a line maps to another point on the line
Work out a invariant line
Suppose for y = mx + c * Multiply matrix and (x y) = (x’ y’) Use * As invariant line y’=mx’ + c Rearrange for m=0 Once solved quadratic for this Put back into x’ and y’ equations Gives invariant lines
Reflection in a plane
(-1 0 0)
(0 1 0)
(0 0 1)
Reflection in plane x=0
Inverse of a 2x2 matrix
M^-1 = 1/(ad-bc) (d -b)
(-c a)
What is it called if detm = 0
Singular
Proof for inverse of a product of matrices
x(mn) = i xmnn^-1 = in^-1 xm = n^-1 xmm^-1 = n^-1m^-1 x = n^-1m^-1
Inverse of a 3x3 matrix
1/det m (inverse)
How can planes are arrange in 3D
Can be parallel either two or three parallel
Can have a unique intersection
Can have infinite solutions then sheaf
No solution, only intersection between each plane, prism
Sum of a geometric series
Sn = a(1-r^n)/1-r
Arithmetic sequence
Common difference denoted by d
Explain two methods of summing series
Partial and differences
Proof of induction
Prove n = 1 Assume true for n = k Let n = k+1 for target expression Use n=k to find n = k+1 True for all values of n
Matrix proof by induction
N=1
A^kA= A^k+1
When is an improper integral convergent and divergent
When a -> infinity , is defined so convergent
What does sin and cos integrate and differentiate too
Remember circle of them
Differentiate y=arcsinx
1/(1-x^2)^0.5
What does tan differentiate too
Sec^2
Cartesian too polar form
X=rcosa
Y=rsina
Area under a polar curve
Integral ( 0.5r^2) dx
Accuracy of a maclaurin expansion
% error = approximate value - exact value / exact value )
Complex conjugate
Opposite of imaginary
Coshx in exponentials
(e^x + e^-x)/2
Sinhx
(e^x - e^-x) / 2
tanhx
(e^x - e^-x) / (e^x + e^-x)
Hyperbolic identities
Cosh^2(x) - sinh^2(x) = 1
Cosh2x = cosh^2(x) + sinh^2(x)
sinh2x = 2sinhxcoshx
What does coshx and sinhx differentiate too
Themselves
What does arsinhx = ?
Ln(x + (x^2 + 1)^0.5)
Artanhx equal too ?
0.5ln( (1 + x) / (1 - x) )
What does arcoshx differentiate too
1/(x^2 - 1)^0.5
Area of a volume
V = integral ( pi y^2 )
Mean of a function f(x)
(1/(b-a))integral (f(x))
What is a root mean square and what’s the formula
How far the function is away from zero on average,,
((1/(b-a)) integral (f(x))^2) ^0.5
Describe simple harmonic motion
D^2x/dt^2 = -w^2 x
Where period is 2pi / w
General solution is Pcoswt + Qsinwt = x
What’s homogenous and non homogeneous
Homogeneous is one side is 0
Explain how the complementary function works
Say dy/dx and d^2/dx^2 are equal too then sub them into differential, divide by Ae^€x to get cf
Name two conditions to solve exact equation of a cf with real distinct roots
Initial conditions or boundary conditions would
Cf for a AE with repeated roots
(A + Bx)e^¥x
Difference between general and particular solution
Particular find constants
How do you choose what trig substitution to do for Further integration
Match the 1+x^2 to for example sec^2 = 1 + tan^2
Modulus argument form
R(cosx + sinx)
What is a loci in form |z-a| = r
Circle
What is the locus of an angle
Arg(z-a) = x
Locus of points on a perpendicular bisector
|z-a| = |z-b|
Explain why the general solution for a second order differential is what it is
Relationship with e^ix
Prove how R works as an integrating factor
Multiply by R
LHS can be written as d/dx(Ry)
Comparing both sides once differentiating
Only true if dR/dx = RP
So integrate both sides
So R = e^integral p dx
Proof for why e^i€ = cos€ + isin€
Maclaurin series for sin€ and cos€ is same as series for e^x when added and is x is replaced with i€
What does cos€ and sin€ equal in complex numbers
Cos€ = 0.5( e^i€ + e^-i€)
Sin€ = 0.5i( e^i€ - e^-i€)
What is z^n and z^-n
Z^n = Cosnx + isinnx
Z^-n = cosnx - isinnx
What does cosnx equal
Z^n + z^-n / 2
Sinnx equal to?
Z^n + z^-n / 2i
What is the PI of the linear function, trig function, polynomial of order n, and exponential function
ax + b
anx^n + an-1x^n-1….
acospx + bsinpx
ae^px
Describe a special case for a non-homogeneous equation
If your equation equals e^3x then use PI of axe^3x
