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Stagione 1
Data di messa in onda
Gen 27, 2016
Linear equations include dy/dt = y, dy/dt = –y, dy/dt = 2ty. The equation dy/dt = y*y is nonlinear.
Linear equations include dy/dt = y, dy/dt = –y, dy/dt = 2ty. The equation dy/dt = y*y is nonlinear.
Data di messa in onda
Gen 27, 2016
The sum rule, product rule, and chain rule produce new derivatives from the derivatives of x^n, sin(x) and e^x. The Fundamental Theorem of Calculus says that the integral inverts the derivative.
The sum rule, product rule, and chain rule produce new derivatives from the derivatives of x^n, sin(x) and e^x. The Fundamental Theorem of Calculus says that the integral inverts the derivative.
Data di messa in onda
Gen 27, 2016
With exponential input, e^st, from outside and exponential growth, e^at, from inside, the solution, y(t), is a combination of two exponentials.
With exponential input, e^st, from outside and exponential growth, e^at, from inside, the solution, y(t), is a combination of two exponentials.
Data di messa in onda
Gen 27, 2016
An oscillating input cos(ωt) produces an oscillating output with the same frequency ω (and a phase shift).
An oscillating input cos(ωt) produces an oscillating output with the same frequency ω (and a phase shift).
Data di messa in onda
Gen 27, 2016
To solve a linear first order equation, multiply each input q(s) by its growth factor and integrate those outputs.
To solve a linear first order equation, multiply each input q(s) by its growth factor and integrate those outputs.
Data di messa in onda
Gen 27, 2016
A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump.
A unit step function jumps from 0 to 1. Its slope is a delta function: zero everywhere except infinite at the jump.
1x7
1.5: Response to Complex Exponential, exp(i*w*t) = cos(w*t)+i*sin(w*t)
Episode overview
Data di messa in onda
Gen 27, 2016
For linear equations, the solution for f = cos(ωt) is the real part of the solution for f = e^iωt. That complex solution has magnitude G (the gain).
For linear equations, the solution for f = cos(ωt) is the real part of the solution for f = e^iωt. That complex solution has magnitude G (the gain).
Data di messa in onda
Gen 27, 2016
The integrating factor e^at multiplies the differential equation, y’=ay+q, to give the derivative of e^-aty: ready for integration.
The integrating factor e^at multiplies the differential equation, y’=ay+q, to give the derivative of e^-aty: ready for integration.
Data di messa in onda
Gen 27, 2016
The integral of a varying interest rate provides the exponent in the growing solution (the bank balance).
The integral of a varying interest rate provides the exponent in the growing solution (the bank balance).
Data di messa in onda
Gen 27, 2016
When –by^2 slows down growth and makes the equation nonlinear, the solution approaches a steady state y(∞) = a/b.
When –by^2 slows down growth and makes the equation nonlinear, the solution approaches a steady state y(∞) = a/b.
Data di messa in onda
Gen 27, 2016
Steady state solutions can be stable or unstable – a simple test decides.
Steady state solutions can be stable or unstable – a simple test decides.
Data di messa in onda
Gen 27, 2016
Separable equations can be solved by two separate integrations, one in t and the other in y. The simplest is dy/dt = y, when dy/y equals dt. Then ln(y) = t + C.
Separable equations can be solved by two separate integrations, one in t and the other in y. The simplest is dy/dt = y, when dy/y equals dt. Then ln(y) = t + C.
Data di messa in onda
Gen 27, 2016
For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency
For the oscillation equation with no damping and no forcing, all solutions share the same natural frequency
Data di messa in onda
Gen 27, 2016
With forcing f = cos(ωt), the particular solution is Y*cos(ωt). But if the forcing frequency equals the natural frequency there is resonance
With forcing f = cos(ωt), the particular solution is Y*cos(ωt). But if the forcing frequency equals the natural frequency there is resonance
Data di messa in onda
Gen 27, 2016
With constant coefficients in a differential equation, the basic solutions are exponentials e^st. The exponent s solves a simple equation such as As^2 + Bs + C = 0.
With constant coefficients in a differential equation, the basic solutions are exponentials e^st. The exponent s solves a simple equation such as As^2 + Bs + C = 0.
Data di messa in onda
Gen 27, 2016
The impulse response g is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.
The impulse response g is the solution when the force is an impulse (a delta function). This also solves a null equation (no force) with a nonzero initial condition.
Data di messa in onda
Gen 27, 2016
Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.
Resonance occurs when the natural frequency matches the forcing frequency — equal exponents from inside and outside.
Data di messa in onda
Gen 27, 2016
A damped forced equation has a particular solution y = G cos(ωt – α). The damping ratio provides insight into the null solutions.
A damped forced equation has a particular solution y = G cos(ωt – α). The damping ratio provides insight into the null solutions.
Data di messa in onda
Gen 27, 2016
Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance).
Current flowing around an RLC loop solves a linear equation with coefficients L (inductance), R (resistance), and 1/C (C = capacitance).
Data di messa in onda
Gen 27, 2016
With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.
With constant coefficients and special forcing terms (powers of t, cosines/sines, exponentials), a particular solution has this same form.
Data di messa in onda
Gen 27, 2016
This method is also successful for forces and solutions such as (at^2 + bt +c) e^st: substitute into the equation to find a, b, c.
This method is also successful for forces and solutions such as (at^2 + bt +c) e^st: substitute into the equation to find a, b, c.
Data di messa in onda
Gen 27, 2016
Combine null solutions y1 and y2 with coefficients c1(t) and c2(t) to find a particular solution for any f(t).
Combine null solutions y1 and y2 with coefficients c1(t) and c2(t) to find a particular solution for any f(t).
Data di messa in onda
Gen 27, 2016
Transform each term in the linear differential equation to create an algebra problem. You can then transform the algebra solution back to the ODE solution, y(t).
Transform each term in the linear differential equation to create an algebra problem. You can then transform the algebra solution back to the ODE solution, y(t).
Data di messa in onda
Gen 27, 2016
The second derivative transforms to s^2Y and the algebra problem involves the transfer function 1/ (As^2 + Bs +C).
The second derivative transforms to s^2Y and the algebra problem involves the transfer function 1/ (As^2 + Bs +C).
Data di messa in onda
Gen 27, 2016
When the force is an impulse δ (t), the impulse response is g(t). When the force is f(t), the response is the “convolution” of f and g.
When the force is an impulse δ (t), the impulse response is g(t). When the force is f(t), the response is the “convolution” of f and g.
Data di messa in onda
Gen 27, 2016
The direction field for dy/dt = f(t,y) has an arrow with slope f at each point t, y. Arrows with the same slope lie along an isocline.
The direction field for dy/dt = f(t,y) has an arrow with slope f at each point t, y. Arrows with the same slope lie along an isocline.
Data di messa in onda
Gen 27, 2016
Solutions to second order equations can approach infinity or zero. Saddle points contain a positive and also a negative exponent or eigenvalue.
Solutions to second order equations can approach infinity or zero. Saddle points contain a positive and also a negative exponent or eigenvalue.
Data di messa in onda
Gen 27, 2016
Imaginary exponents with pure oscillation provide a “center” in the phase plane. The point (y, dy/dt) travels forever around an ellipse.
Imaginary exponents with pure oscillation provide a “center” in the phase plane. The point (y, dy/dt) travels forever around an ellipse.
Data di messa in onda
Gen 27, 2016
A second order equation gives two first order equations for y and dy/dt. The matrix becomes a companion matrix.
A second order equation gives two first order equations for y and dy/dt. The matrix becomes a companion matrix.
Data di messa in onda
Gen 27, 2016
A critical point is a constant solution Y to the differential equation y’ = f(y). Near that Y, the sign of df/dy decides stability or instability.
A critical point is a constant solution Y to the differential equation y’ = f(y). Near that Y, the sign of df/dy decides stability or instability.
Data di messa in onda
Gen 27, 2016
With two equations, a critical point has f(Y,Z) = 0 and g(Y,Z) = 0. Near those constant solutions, the two linearized equations use the 2 by 2 matrix of partial derivatives of f and g.
With two equations, a critical point has f(Y,Z) = 0 and g(Y,Z) = 0. Near those constant solutions, the two linearized equations use the 2 by 2 matrix of partial derivatives of f and g.
Data di messa in onda
Gen 27, 2016
Two equations y’ = Ay are stable (solutions approach zero) when the trace of A is negative and the determinant is positive.
Two equations y’ = Ay are stable (solutions approach zero) when the trace of A is negative and the determinant is positive.
Data di messa in onda
Gen 27, 2016
A box in the air can rotate around its shortest and longest axes. Around the middle axis it tumbles wildly.
A box in the air can rotate around its shortest and longest axes. Around the middle axis it tumbles wildly.
Data di messa in onda
Gen 27, 2016
An m by n matrix A has n columns each in Rm. Capturing all combinations Av of these columns gives the column space – a subspace of Rm.
An m by n matrix A has n columns each in Rm. Capturing all combinations Av of these columns gives the column space – a subspace of Rm.
Data di messa in onda
Gen 27, 2016
Vectors v1 to vd are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension d = number of basis vectors.
Vectors v1 to vd are a basis for a subspace if their combinations span the whole subspace and are independent: no basis vector is a combination of the others. Dimension d = number of basis vectors.
Data di messa in onda
Gen 27, 2016
A matrix produces four subspaces – column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.
A matrix produces four subspaces – column space, row space (same dimension), the space of vectors perpendicular to all rows (the nullspace), and the space of vectors perpendicular to all columns.
Data di messa in onda
Gen 27, 2016
A graph has n nodes connected by m edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.
A graph has n nodes connected by m edges (other edges can be missing). This is a useful model for the Internet, the brain, pipeline systems, and much more.
Data di messa in onda
Gen 27, 2016
The incidence matrix A has a row for every edge, containing -1 and +1 to show the two nodes (two columns of A) that are connected by that edge.
The incidence matrix A has a row for every edge, containing -1 and +1 to show the two nodes (two columns of A) that are connected by that edge.
Data di messa in onda
Gen 27, 2016
The eigenvectors x remain in the same direction when multiplied by the matrix (Ax = λx). An n x n matrix has n eigenvalues.
The eigenvectors x remain in the same direction when multiplied by the matrix (Ax = λx). An n x n matrix has n eigenvalues.
Data di messa in onda
Gen 27, 2016
A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λis the eigenvalue matrix.
A matrix can be diagonalized if it has n independent eigenvectors. The diagonal matrix Λis the eigenvalue matrix.
Data di messa in onda
Gen 27, 2016
Diagonalizing A = VΛV–1 also diagonalizes An = VΛnV–1.
Diagonalizing A = VΛV–1 also diagonalizes An = VΛnV–1.
Data di messa in onda
Gen 27, 2016
dy/dt = Ay contains solutions y = eλtx where λ and x are an eigenvalue / eigenvector pair for A.
dy/dt = Ay contains solutions y = eλtx where λ and x are an eigenvalue / eigenvector pair for A.
Data di messa in onda
Gen 27, 2016
The shortest form of the solution uses the matrix exponential y = eAt y(0). The matrix eAt has eigenvalues eλt and the eigenvectors of A.
The shortest form of the solution uses the matrix exponential y = eAt y(0). The matrix eAt has eigenvalues eλt and the eigenvectors of A.
Data di messa in onda
Gen 27, 2016
A and B are “similar” if B = M-1AM for some matrix M. B then has the same eigenvalues as A.
A and B are “similar” if B = M-1AM for some matrix M. B then has the same eigenvalues as A.
1x45
6.5: Symmetric Matrices, Real Eigenvalues, Orthogonal Eigenvectors
Episode overview
Data di messa in onda
Gen 27, 2016
Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.
Symmetric matrices have n perpendicular eigenvectors and n real eigenvalues.
Data di messa in onda
Gen 27, 2016
An oscillation equation d2y/dt2 + Sy = 0 has 2n solutions (sines and cosines). Solutions use the eigenvectors of S.
An oscillation equation d2y/dt2 + Sy = 0 has 2n solutions (sines and cosines). Solutions use the eigenvectors of S.
Data di messa in onda
Gen 27, 2016
A positive definite matrix S has positive eigenvalues, positive pivots, positive determinants, and positive energy vTSv for every vector v. S = ATA is always positive definite if A has independent columns.
A positive definite matrix S has positive eigenvalues, positive pivots, positive determinants, and positive energy vTSv for every vector v. S = ATA is always positive definite if A has independent columns.
Data di messa in onda
Gen 27, 2016
The SVD factors each matrix A into an orthogonal matrix U times a diagonal matrix Σ (the singular value) times another orthogonal matrix VT: rotation times stretch times rotation.
The SVD factors each matrix A into an orthogonal matrix U times a diagonal matrix Σ (the singular value) times another orthogonal matrix VT: rotation times stretch times rotation.
Data di messa in onda
Gen 27, 2016
A second order equation can change its initial conditions on y(0) and dy/dt(0) to boundary conditions on y(0) and y(1).
A second order equation can change its initial conditions on y(0) and dy/dt(0) to boundary conditions on y(0) and y(1).
Data di messa in onda
Gen 27, 2016
The partial differential equation ∂2u/∂x2 + ∂2u/∂y2 = 0 describes temperature distribution inside a circle or a square or any plane region.
The partial differential equation ∂2u/∂x2 + ∂2u/∂y2 = 0 describes temperature distribution inside a circle or a square or any plane region.
Data di messa in onda
Gen 27, 2016
A Fourier series separates a periodic function F(x) into a combination (infinite) of all basis functions cos(nx) and sin(nx).
A Fourier series separates a periodic function F(x) into a combination (infinite) of all basis functions cos(nx) and sin(nx).
Data di messa in onda
Gen 27, 2016
Even functions use only cosines (F(–x) = F(x)) and odd functions use only sines. The coefficients an and bn come from integrals of F(x) cos(nx) and F(x) sin(nx).
Even functions use only cosines (F(–x) = F(x)) and odd functions use only sines. The coefficients an and bn come from integrals of F(x) cos(nx) and F(x) sin(nx).
Data di messa in onda
Gen 27, 2016
Inside a circle, the solution u(r, θ) combines rn cos(nθ) and rn sin(nθ). The boundary solution combines all entries in a Fourier series to match the boundary conditions.
Inside a circle, the solution u(r, θ) combines rn cos(nθ) and rn sin(nθ). The boundary solution combines all entries in a Fourier series to match the boundary conditions.
Data di messa in onda
Gen 27, 2016
The heat equation ∂u/∂t = ∂2u/∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.
The heat equation ∂u/∂t = ∂2u/∂x2 starts
.. show full overview
The heat equation ∂u/∂t = ∂2u/∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.
The heat equation ∂u/∂t = ∂2u/∂x2 starts from a temperature distribution u at t = 0 and follows it for t > 0 as it quickly becomes smooth.
Data di messa in onda
Gen 27, 2016
The wave equation ∂2u/∂t2 = ∂2u/∂x2 shows how waves move along the x axis, starting from a wave shape u(0) and its velocity ∂u/∂t(0).
The wave equation ∂2u/∂t2 = ∂2u/∂x2 shows how waves move along the x axis, starting from a wave shape u(0) and its velocity ∂u/∂t(0).
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