Pure Flashcards by Huwbigboy jones

Critical damping

Discriminant of AE is 0

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Underdamping

Discriminant is negative

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Overdamping

Discriminant is positive

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Scaler product

a.b=|a|.|b|cosx

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Vector equation of a plane

(r-a).n=0

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Cartesian equation of a plane

n1x+n2y+n3z+d=0

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Find angle between two planes

Use scalar product on normals

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Vector equation of a line in 2d

R=a+¥d

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Cartesian equation of line 2d same for 3d

¥= x-a/d=y-a1/d1

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Special case for Cartesian equation of a line in 3D

When d=0 write as x=a , ¥=y-a1/d1

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Intersection of two lines in 2d

Solve direction simultaneously

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How to prove to lines in 2d are skew

No unique solution simultaneously

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Enlargement scale factor k about origin

(K 0)

O k

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Stretch scale factor m parallel to x axis

(m 0)

0 1

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Stretch scale factor n parallel to y axis

(1 0)

0 n

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Rotation 90 degrees clockwise about origin and any rotation about origin through x degrees

(0 1)
(-1 0)

(Cosx -sinx)
(Sinx cosx)

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Reflection in x and y axis

X axis (1 0)
           (0 -1)

Y axis (-1 0)
           (0 1)

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Reflection y=x

(0 1)

1 0

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Equation of a shear

Shear x axis fixed (1 k)
(0 1)

Y axis fixed (1 0)
(k 1)

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How do you rotate 3D shapes

Check book if unsure

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What is an invariant point

A point that doesn’t change when multiplied by a matrix

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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

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What is an invariant line

Any point of a line maps to another point on the line

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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

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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

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