Captured On [2020-01-21 Tue 13:07] Source Linear Systems | Unit IV First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 Can we solve a linear system of ODE’s with constant coefficients by eliminating variables? 1.1 Front Can we solve a linear system of ODE’s with constant coefficients by eliminating variables? 1.2 Back No, it’s a naive way to solve it. You need to use techniques of constant coefficient ODE methods....
Captured On [2020-02-01 Sat 12:35] Source Matrix Methods: Eigenvalues and Normal Modes | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the trace of a square matrix? 1.1 Front What is the trace of a square matrix? For example, \({\displaystyle \operatorname{tr}\begin{pmatrix}a & b \\ c & d\end{pmatrix}}\) 1.2 Back It’s the sum of the elements on the main diagonal; it’s denoted \(\operatorname{tr}(A)\):...
Captured On [2020-02-13 Thu 17:57] Source Qualitative Behavior: Phase Portraits | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the phase plane for this system? 1.1 Front What is the phase plane for this system? \({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \({\displaystyle A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}}\) 1.2 Back It’s the \(xy\text{-plane}\) itself, where you can draw the trajectory of a solution with an arrow to indicate the direction of increasing time....
Captured On [2020-02-20 Thu 21:51] Source Matrix Exponentials | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 For this linear system, how many solutions are there? 1.1 Front For this linear system, how many solutions are there? \({\displaystyle \dot{\vb{x}} = A \vb{x}}\), where \(A\) is a \(n \cross n\) matrix 1.2 Back There are \(n\) linearly independent solutions for the system 2 How is the linear system when the coefficients are functions of the independent variable t?...
Captured On [2020-03-06 Fri 14:04] Source Nonlinear Systems | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is an autonomous system? 1.1 Front What is an autonomous system? General First Order Autonomous Systems 1.2 Back The word autonomous means self regulating. These systems are self regulating in the sense that their rate of change (e.g. derivatives) depends only on the state of the system (values of \(x\) and \(y\)) and not on the time \(t\)...
Captured On [2020-03-09 Mon 12:02] Source Linearization Near Critical Points | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 How can we get information about the trajectories of autonomous linear system without… 1.1 Front How can we get information about the trajectories of autonomous linear system without… determining them analytically or using a computer to plot them 1.2 Back Analyzing what happens near critical points....
Captured On [2020-03-16 Mon 19:13] Source Limitations of the Linear: Limit Cycles and Chaos | Unit IV: First-order Systems | Differential Equations | Mathematics | MIT OpenCourseWare 1 What is the form of the solutions that trace out a closed curve? 1.1 Front What is the form of the solutions that trace out a closed curve? In a non-linear system 1.2 Back The solution \(\vb{x}(t)\) will be geometrically realized by a point which goes round and round the curve \(C\) with a certain period \(T\)...
We start this unit by learning to visualize functions of several variables using graphs and level curves. Following this we will study partial derivatives. These will be used in the tangent approximation formula, which is one of the keys to multivariable calculus. It ties together the geometric and algebraic sides of the subject and is the higher dimensional analog of the equation for the tangent line found in single variable calculus. We will use it in part B to develop the chain rule. We will apply our understanding of partial derivatives to solving unconstrained optimization problems. (In part C we will solve constrained optimization problems.) Last Modification: 2020-02-07 Fri 16:44 ...
Captured On [2020-02-05 Wed 19:40] Source Session 1: Vectors | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Definition of vectors Figure 2: Length of a vector Figure 3: Modules and addition Figure 4: Multiplying by scalars 2 Which is the vector between 2 points? 2.1 Front Which is the vector between 2 points?...
Captured On [2020-02-05 Wed 21:18] Source Session 10: Meaning of Matrix Multiplication | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What represent matrix multiplication Figure 2: Matrix identity and plane rotation by matrix multiplication Figure 3: Continuous apply of rotation by matrix multiplication 2 Can we use the distributive law with matrix multiplication?...
Captured On [2020-02-05 Wed 21:19] Source Session 11: Matrix Inverses | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Inverse Matrix Figure 2: Inverse Matrix formula Figure 3: Inverse Matrix Formula: Minors and cofactors Figure 4: Inverse Matrix Formula: Transpose and Divide by \(\det(A)\) 2 How can we solve a small squared linear system?...
Captured On [2020-02-05 Wed 21:20] Source Session 12: Equations of Planes II | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Equations of Planes Figure 2: Point in the plane Figure 3: Normal vector and vectors on the plane Figure 4: Extract normal vector from plane equation Figure 5: Check vector parallel or perpendicular to a plane...
Captured On [2020-02-05 Wed 21:21] Source Session 13: Linear Systems and Planes | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: 3x3 Linear System Figure 2: Possible solutions to linear systems (2) 2 How many solutions we can get from a linear system? 2.1 Front How many solutions we can get from a linear system?...
Captured On [2020-02-05 Wed 21:22] Source Session 14: Solutions to Square Systems | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Trivial solution and inverse of a matrix Figure 2: Solutions of homogeneous linear system Figure 3: Coplanar normal vectors Figure 4: General case of solutions 2 When can we say that a linear system has an unique solution?...
Captured On [2020-02-05 Wed 21:25] Source Session 15: Equations of Lines | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Equations of lines Figure 2: Example of line through 2 points Figure 3: Position at time \(t\) Figure 4: Get the parametric lines through 2 points 2 How does work a parametric curve? 2.1 Front How does work a parametric curve?...
Captured On [2020-02-05 Wed 21:25] Source Session 16: Intersection of a Line and a Plane | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Application: Intersection with a plane Figure 2: Where are the points respect to the plane? Figure 3: When will be an intersection to the plane? Figure 4: What does happen when the line is parallel to the plane or in the plane...
Captured On [2020-02-05 Wed 21:25] Source Session 17: General Parametric Equations; the Cycloid | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Parametric equations for arbitrary motion (plane/space) Figure 2: Cycloid path movement Figure 3: Which is the position \((x(\theta), y (\theta))\) Figure 4: Vector position \(\vec{OP}\) Figure 5: Getting vectors Figure 6: Vector position \(\vec{OP}\)...
Captured On [2020-02-05 Wed 21:26] Source Session 18: Point (Cusp) on Cycloid | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What happens near bottom? Figure 2: Get an approximation for \(\theta\) small Figure 3: Taylor approximations for \(\sin\) and \(\cos\) when \(\theta\) is small Figure 4: Slope when \(\theta\) is small 2 How can we analyze what happens at cusps on a cycloid graph?...
Captured On [2020-02-05 Wed 21:26] Source Session 19: Velocity and Acceleration | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review parametric equations and position vector Figure 2: Velocity in a cycloid Figure 3: Calculating module of speed Figure 4: Acceleration a vector, and warning about derivative of a module 2 How can we get the velocity vector from a position vector?...
Captured On [2020-02-05 Wed 19:41] Source Session 2: Dot Products | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Dot Product Definition Figure 2: What does geometic definition mean? Figure 3: Dot product of combined vectors 2 What is the dot product of 2 vectors? 2.1 Front What is the dot product of 2 vectors?...