Math Flashcards
What is Bernoulli’s equation to solve nonlinear ODES
In the form: y’+p(x)y=g(x)y^a
Y^1-a=u
u’+(1-a)pu=(1-a)g
What are the three responses of a damped spring system and what are the corresponding eigenvalues?
- Critically Damped: real double root
- Underdamped: complex root
- Overdamped: distinct real roots
What is the Euler Cauchy equation?
x^2y’’+axy’+by=0
y=x^m
m^2+(a-1)m+b=0
- Y=cx^m
- Y=(c1+c2lnx)x^m
- Y=x^alpha *[Acos(wlnx)+Bsin(wlnx)]
How do you represent the particular solution portion of a forced oscillation?
r(t)=F_ocos(wt)
What are the five kinds of critical points of a system of equation?
- Proper node (Real Same Sign wjth vectors of x and y axis)
- Improper node (Real Same Sign)
- Saddle point (real opposite sign)
- Center (pure imaginary)
5 spiral point (complex)
What are the stability rules for a system of equations?
- Stable and attractive: both negative and non zero
- Stable: both negative or 0
- Unstable: either positive
What are the steps of finding a nonhomogenous solution for a system of equations?
- Solve homogenous solution. Obtain Y(t)c
- Use that Y(t) within u’(t)=Y^-1g (g is the nonhomogenous portion of the result)
- Find Y^-1 and multiply by g
- Take the integral of u’(t)
5 multiple Y(t) by u(t) and add it to the homogenous solution.
What is the generic power series(do not include the series symbol)?
A_m(x-x_0)^m
What is the Fourier power series for
- e^x
- 1/1-x
- cos(x)
- Sin(x)
- X^m/m!
- x^m
- (-1)^m(x^2m)/(2m!)
- (-1)^m(x^2m+1)/(2m+1!)
What is the integral for Laplace Transforms?
L(f)=integral from 0 to inf (e^-st)*f(t)dt
What is the relationship for s shifting?
L[e^at*f(t)]=F(s-a)
What is the relationship for taking the Laplace transform of an integral?
L{integral from 0 to t f(Tau)dTau=F(s)/s
Explain what a unit step function is and how the laplace transform look.
Apply it next to a function to ‘activate’ that function at the given time a. Applying it with a minus sign with ‘deactivate’ that function.
L{u(t-a)}=(e^-as)/s
What is the t-shifting laplace transform?
L{f(t-a)u(t-a)}=(e^-as)*F(s)
-or-
L{f(t)u(t-a)}=(e^-as)L{f(t-a)}
What is the laplace transform for a short impulse?
L{delta(t-a)}=e^-as
How do you use convolution?
Given L(fg)=H(s)
h(t)=(fg)(t)=integral from 0 to t f(Tau)g(t-Tau)dTau
How to use the Variation of Parameters to solve a second order nonhomogenous ODE
Find homogenous solutions and use generic solutions as y1 and y2
yp(x)=-y1integral((y2r)/W)+y2integral((y1r)/W)
W=y1y2’-y1’y2
What is the inverse of a 3x3 matrix
[ c11 -c21 c31
1/det(A) * |-c12 c22 -c32|
[c13 -c23 c33]
What distinguishes a 1. Symmetric Matrix 2. Skew-Symmetric Matrix 3. Orthogonal Matrix
- A=A^T (Real Eigenvalues)
- A=-A^T (Pure Imaginary or Zero Eigvenvalues)
- A^-1=A^T (Pure Imaginary or Zero Eigvenvalues)
What is a diagonalized matrix?
A^^=P^-1AP
A^^=[Eigen1 0;0 Eigven2]
P=[v1 v2];
D^M=x^-1A^MX
What is the inner product? Include the angular relationship between the two vectors.
a*b=a1b1+a2b2+a3b3
cos(gamma)=a*b/|a||b|
What is an example of an inner product?
Wrok done by a force W=p*d
What is the equation for a projection of Vector A in the direction of Vector B?
P=a*b/|b|
What is the Vector product? Include the angular relationship between the two vectors.
Take the cross product of vector A and B
sin(gamma)=|axb|/|a||b|
What are examples of cross product use?
Moment of a force. M=rxp
What is the sclar triple product?
a*(bxc)
What are the parametric representations of
- Circle
- Ellipse
- Straight Line
- Circular Helix
- r(t)=[acost,asint,0]
- r(t)=[acost,bsint,0]
- r(t)=[a1+b1t,a2+b2t,a3+tb3]
- r(t)=[acost,bcost,ct]
What is the equation for the length of a curve?
l=integral from a to b of (sqrt(r’*r’)dt)
What is the equation for arc length?
s(t)= integral from a to t of (sqrt(r’*r’)dt)
What is the equation for the tangent of a curve?
q(w)=r+r’(w)
What is the equation for the curvature of a line?
K(s)=|u’(s)|=|r’‘(s)|
What is the equation for the torsion of a line?
Tau(s)=|b’(s)| where b(s)=u x u’/K
What is the definition of the gradient?
Rate of change of a scalar field f(x,y,z) in a given direction in space. It points in the direction of maximum increase of f
What is the equation of the gradient
grad(f)=[df/dx,df/dy,df/dz]
What is the equation for the directional derivative?
D_bf=b/|b| *grad(f) where b is the direction
What is the Laplace Operator?
Triangle or Upside Triangle Square = [d^2/dx^2,d^2/dy^2,d^2/dz^2]
div(gradf)
What is the divergence?
How much flow is expanding/contracting at a given point?
What is the equation for the divergence?
div V= v1/dx+dv2/dy+dv3/dz
What does it mean to be incompressible?
div V=0
What is the curl of a vector?
The tendency of a vector to rotate about a given point
What is the equation for the curl?
The cross product of the Differential Operator and v
What does it mean to be rotational/irrotational?
Irroational: curl f=0
How do you check if a function is conservative or not?
Nonconservative: curl F ~=0
Conservative: Find the potential function such that grad f=F
What is the equation for a line integral?
int of c (F(r)dr)= int from a to b(F(r(t))r’(t) dt)
How can you check for path independence/exactness?
- Find F=grad(f)
2. curl F=0
How can you simplify the calculation of a line integral if you know it is path independent?
First from F(t) find f(t)
int from a to b(F[r(t)]*r’(t))= f(B)-f(A)
What is the transition from cartesian to radial coordinates for a double integral?
int int f(x,y) dx dy= int int f(rcos(theta),rsin(theta))r dr dtheta
What does Green’s Theorem accomplish?
It provides a transformation between a double integral and a line integral
What is the equation for Green’s Theorem?
int int (dF2/dx-dF1/dy) dx dy= int F*dr
What is the equation for finding the area of an enclosed line?
A=1/2 int xdy-ydx
What is the parametric representation of a
- Cylinder
- Sphere
- Cone
- r(u,v)=[acosu,asinu,v]
- r(u,v)=[acosucosv,asinucosv,sinv]
- r(u,v)=[ucosv,usinv,u]
Find the unit normal vector given representation g(x,y,z)
n=grad(g)/|grad(g)|
What is the equation for a surface integral?
int int F_n Da= int int F(r(u,v))*N(u,v) du dv
where N(u,v)=r_u x r_v
What is the equation for the area of a surface?
A(s)=int int |r_u x r_v| du dv
What does the Divergence Theorem of Gauss do?
Provides a transformation between triple integrals and surface integrals
What is the Divergence Theorem of Gauss Eq?
int int int div(F) dV= int int F*n dA
What does Stoke’s Theorem accomplish?
It provides a transformation between surface and line integral
What is the Stoke’s Theorem eq?
int int (curl F)n DA=int F(r(t)r’(t) ds
What do Fourier Series do?
They are infinite series that represent periodic functions in terms of cosines and sines.
What is the equation and main components of a basic Fourier series?
f(x)=a_0 + Summation n=1 to inf(a_n cos(nx)+b_n sin(nx))
a_0=1/2pi int from -pi to pi (f(x)) dx
a_n= 1/pi int from -pi to pi (f(x) cosnx) dx
b_n 1/pi int -pi to pi (f(xO)sin(nx) dx
What are half range expansions?
expanding a fourier series to be either even or odd depending on how you treat a_n and b_n
What is the Fourier series equation for an arbitrary period?
L= period/2
f(x)=a_0 +summation(n=1 to inf) a_n cos(npix/L) + b_n sin(n pi x/L)
a_0= 1/2L int(-L to L) (f(x) dx
a_n=1/L int(-L to L) f(x) cos(n pi x/L) dx
b_n1/L int(-L to L)f(x) sin(n pi x/L) dx
What happens to a Fourier series for an odd function?
A_0 and a_1= 0
What happens to a Fourier series for an even function?
b_n=0
What is the error equation to determine the error in a Fourier series approximation?
E=int (-pi to pi) (f^2 dx)- pi(2a_0^2+summation (n=1 to N) a_n^2+b_n^2)
What is the equation to determine if two functions are orthogonal?
(y_m,y_n)=int (from a to b) r(x)y_m(x)y_n(x)dx=0
What is the form of the Sturm-Liouville equation?
[p(x)y’]’+[q(x)+r(x)llamba]y=0
What is the equation for the Fourier Integral?
f(x)=int(from 0 to inf) [A(w)cos(wx)+B(w)sin(wx) dw]
A(w)=1/pi int(-inf to inf) (f(v)cos(wv)dv
B(w)=1/pi int(-inf to inf) (f(v)sin(wv) dv
What is the Cosine Fourier Integral equation?
For an even function
f(x)=integral (0 to inf) (A(w)cos(wx) dw
A(w) 2/pi integral(0 to inf) (f(v) cos(wv) dv
What is the sine fourier integral?
f(x)= integral(0 to inf) B(w)sin(wx) dw
B(w)= 2/pi int(0 to inf) (f(v)sin(wv) dv
What is an integral transform?
A transformation in the form of an integral that produces from given functions new functions depending on a different variable
What is the equation for the cosine fourier transform?
f_c(w)=sqrt(2/pi) int (0 to inf) f(x) cos(wx) dx
What is the equation for the sine fourier transform?
f_s(w)=sqrt(2/pi) int(o to inf) f(x) sin(wx) dx
What is the fourier transform?
f(w)=1/sqrt(2pi) int(o to inf) f(x) e^-iwx dx
What is the 1D wave equation?
d^2u/dt^2=c^2*(d^2u/dx^2)
What is the 1D heat equation?
du/dt=c^2*(d^2u/dx^2)
What is the 2D Laplace equation?
d^2u/dx^2+d^2u/dy^2=0
What is the 2D Poisson Equation?
d^2u/dx^2+d^2u/dy^2=f(x,y)
What is the 2D Wave Equation?
d^2u/dt^2=c^2(d^2u/dx^2+d^2u/dy^2)
What is the 3D Laplace Equation?
d^2u/dx^2+d^2u/dy^2+d^2u/dz^2=0
What are the three steps to finding the solution to a PDE?
- Separation of Variables
- Determine Solutions of the ODES that satisfy the BCs
- Use Fourier Series to Compose Solutions found in step 2.
What is the first step for the Separation of Variables?
u=F(x)G(y)
Take the partial derivatives
What are the two relationships that can be found using complex conjugate relating to the real and imaginary portions of the function, z.
- Re(z)=x=(1/2)(z+z_conj)
2. Im(z)=y=(1/2i)(z-z_conj)
What is the triangle inequality?
|z1+z2|<= |z1|+|z2|
What is the equation for the multiplication of complex functions in polar form?
z1z2=r1r2(cos(theta1+theta2)+isin(theta1+theta2)
What is the equation for the division of complex functions in polar form?
z1/z2=r1/r2(cos(theta1-theta2)+isin(theta1-theta2))
What is the argument (arg) of a complex function?
The angle between the function and the x-axis.
What do the Cauchy-Riemann Equations do for complex functions?
Check to see if they are analytic on some domain and therefore has a derivative
What are the Cauchy-Riemann Equations?
u_x=v_y
u_y=-v_x
What are the two equations for the derivative of a complex function f(z)?
- f’(z)=u_x+iv_x
2. f’(z)=-iu_y+v_y
What is the equation for a complex exponent?
e^z=e^x(cos(y)+isin(y))
What is Euler’s formula for complex exponents?
e^z=e^iy=cos(y)+isin(y)
x=0
Show that the polar form of a complex number can be represented using Euler’s formula.
z=r*e^i(Theta)=r(cos(theta)+isin(Theta))
How do you find z given a trig function cos(z)
- Set up the trig representation
- Multiple by e^iz
- Form a quadratic equation and find y
- Set e^iz=y and solve for z
What is the natural log equation for a complex function?
ln(z)=ln(r)+i Theta
What is the equation for a complex function raised to a constant power?
z^c=e^cln(z)
What is the equation for a constant raised to the power of a complex function?
a^z=e^zln(a)
What is the equation for
- cos(z)
- sin(z)
- cos(z)=(1/2)(e^iz+e^iz)
2. sin(z)=1/2i*(e^iz-e^-iz)
What is the equation to represent the integral of a complex function along a path?
int(f(z) dz=int(from a to b) (f(z(t))*z’(t) dt
What is Cauchy’s Integral theorem?
If f(z) is analytical and the integration is along a simply connected closed path
int_c(f(z) dz=0
What is Cauchy’s integral equation?
int_c(f(z)/z-z0 dz)= 2pii*f(z0)
What is the equation for the derivative of a complex analytical function?
int_c (f(z)/(z-z0)^n+1)=(2pii*f^n(z0))/n!
What is the test to see if a series has converged?
Lim n-> inf z_n= C where c DNE infinity
What does it mean for a series to be absolutely convergent?
The series of the absolute value of z_n converges
What is the series ratio test?
If |z_n+1|/|z_n|<=q<1 The series converges
or set L=lim n to inf |z_n+1|/|z_n|
Otherwise, it diverges
What is the series root test?
If (z_n)^1/n<=q<1 Converges
otherwise Diverges
How do you calculate the radius of convergence of a series?
R=1/L
L=lim n->inf |a_n+1|/|a_n|
What is the Taylor series?
f(x)=summation from (0 to inf) f^n(x) (x-xo)^n/n!
What is the Maclaurin Series?
f(x)= summation from (o to inf) f^n(x0) (x)^n/n!
What is the equation for a Mobius Transform?
w=(az+b/cz+d)
What is the equation for an Inverse Mobius Transform?
z=(dw-b/-cw+a)
What is the special linear fractional transform equation?
(w-w1) (w2-w3) (z-z1) (z2-z3)
——– * ———— = ——– * ———–
(w-w3) (w2-w1) (z-z3) (z2-z1)
What is newtons method for numerical analysis?
x_n+1=x_n-(f(x_n)/f’(x_n))
What is the secant method for numerical analysis?
x_n+1=x_n-f(x_n)*[(x_n+x_n-1/f(x_n)+f(x_n-1))]
What is the Weierstrass Approximation theorem?
A theorem used to describe that for any continuous function f(x) on an interval [a,b] and error bound Beta>0, there is a polynomial p_n(x) such that
|f(x)-p_n(x)|
What is the equation for a linear Lagrange interpolation?
p1(x)=L0f0+L1f1
L0=(x-x1)/(x0-x1)
L1=(x-x0)/(x1-x0)
What is the equation for a quadratic Lagrange interpolation?
p1=L0f0+L1f1+L2f2
L0=[(x-x1)(x-x2)/(x0-x1)(x0-x2)]
L1=[(x-x0)(x-x2)/(x1-x0)(x1-x2)]
L2=[(x-x0)(x-x1)/(x2-x0)(x2-x1)]
What is the equation for Newton’s Forward Difference Formula?
f(x) = p_n(x)= Summation from s=0 to n (r Detla^s f0
s)
where Delta^s f0= Delta^s-1 f1 - Delta^s-1 f0
What is the equation for Newton’s Backward Difference Formula?
f(x0=p_n(x)=Sum from s=o to n (r+n-1 Del^s f0
s)
Del^s f0= Del^s-1 f0 Del^s-1f-1
What is the rectangular rule for numeric integration?
J=(b-a)/n[f(x1)+f(x2)+f(x3)+….+f(xn)]
What is the trapezoid rule for numeric integration?
J=(b-a)/n[f(a)/2 +f(x1)+f(x2)+…+f(xn-1)+1/2f(b)]
How do you calculate the error of the trapezoid rule?
e=-((b-a)h^2f’‘(t))/12
What is the equation for Simpson’s rule of integration?
J=h/3[f0+4f1+2f2+….+2f2m-2+4f2m-1+f2m]
h=b-a/2m
How do you calculate the error of Simpson’s rule?
e=-(b-a)/180 h^(4)f’’’‘(t)
What is the approximation equation for numeric integration?
f’(x)=(2x-x1-x2) (2x-x0-x2) (2x-x1-x2)
———— f0 - ————– f1 + ————— f2
2h^2 h^2 2h^2
What is the equation for LU-Factorization (Doolittle Method)?
A=LU
Ly=b
Ux=y
L: Lower Triangular Matrix with identity on the diagonal
U: Upper Triangular Matrix
What is the Cholesky’s Method?
A=L*L^T
Ly=b
L^Tx=y
What is the difference between direct methods and indirect methods of solving linear systems of equations?
Direction methods give solutions after an amount of computation that is specified in advance
Indirection methods start from an approximation and obtained better and better approximations as the number of iterations increases.
How does the Gauss-Seidel Iteration Method Work?
- Isolate x1,x2…xn on one side.
- Input an initial guess for x
- As you determine that values for x1, x2… use those new values on the same iteration step.
- Continue until an appropriate level of accuracy has been reached
How do you test if a system will converge using the Gauss-Seidel Iteration method?
Understand that A=I+L+U
C=-(I+L)^-1*U
if ||C||<1 then the system converges
What is the difference between simultaneous corrections and successive corrections?
Simultaneous correction (Jacobi): No new component of x^m is used until all the components of x^m have been computed
Successive Corrections (Gauss -Seidel): An approximation for an individual term within x^m is updated as soon as it is calculated instead of waiting for all components of x^m to be calculated
Does it mean if a matrix is ill-conditioned?
A small change in the input leads to a large change in the output
How can you tell if a matrix is well-conditioned?
If the main diagonal entries of A have large absolute values compared to those of the other entries. Also, the condition number is small.
What are the three types of vector norms?
- l1 norm ||x||_1= |x1|+|x2|+…+|xn|
- l2 norm/ Euclidean ||x||_2=sqrt(x1^2+x2^2+x3^2+…+xn^2)
- i_inf
||x||_inf=max(|xj|)
How do you find the norm of a matrix?
||A||_1=max of summation of absolute value of column components
||A||_inf= max of summation of absolute value of row components
How do you find the condition number of a matrix?
K(A)=||A||*||A^-1||
What is the equation for the least-squares method to find the equation of a straight line?
y=a+bx
an+bSum(x)=sum(y)
aSum(x)+bSum(x^2)=Sum(x*y)
How can you calculate the error of an iterative solution?
Find the residual
r=b-aX^n
How do you find the distance traveled by a particle over time?
take the integral over time of the norm of the velocity
WHat is the equation for finding the centroid of a surface?
x_hat=int(x_tilda*dA)/ int(dA) x_tilda=half of the x value within the interval dx (For vertical analysis this is x)
y_hat=int(y_tilda*dA)/ int(dA) y_tilda=half of the y value within the interval dx (For horizontal analysis this is y)
What is Cramer’s rule and how can you tell if you can use it?
x=D/x_D
y=D/y_D
You must ensure D dne 0.
Where x_D and y_D are the determinant when you replace the x or y values with the B values respectively.
What is the Reimann’s Sum
The summation used to define the area under a curve as the integral
What is the Taylor series expansion for linearization?
f(x)=f(x0)+df(x0)/dx*Delta(X)
Continue for as many variables you have for f(x)
