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.
1x3 1.4b: Response to Exponential Input, exp(s*t) Episode overview
Air date
Jan 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.
1x4 1.4c: Response to Oscillating Input, cos(w*t) Episode overview
Air date
Jan 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).
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.
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.
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.
1x17 2.4: Exponential Response - Possible Resonance Episode overview
Air date
Jan 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.
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).
1x23 2.7: Laplace Transform: First Order Equation Episode overview
Air date
Jan 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).
1x24 2.7b: Laplace Transform: Second Order Equation Episode overview
Air date
Jan 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).
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.
1x31 3.3b: Linearization of y'=f(y,z) and z'=g(y,z) Episode overview
Air date
Jan 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.
1x32 3.3c: Eigenvalues and Stability: 2 by 2 Matrix, A Episode overview
Air date
Jan 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.
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.
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.
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.
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.
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.
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).
1x53 8.1c: Fourier Series Solution of Laplace's Equation Episode overview
Air date
Jan 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.
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.
