Captured On [2020-02-05 Wed 21:28] Source Session 20: Velocity and Arc Length | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Arc lenght Figure 2: Lenght of an arch of cycloid Figure 3: Velocity vector Figure 4: Limit as \(\Delta t \to 0\) 2 What does means speed? 2.1 Front What does means speed?...
Captured On [2020-02-05 Wed 21:29] Source Session 21: Kepler’s Second Law | Part C: Parametric Equations for Curves | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Description of kepler’s second law Figure 2: Kepler’s law in terms of vectors? Figure 3: Area for second law Figure 4: Motion of the plane in the same plane Figure 5: Implications of Kepler’s second law...
Captured On [2020-02-06 Thu 17:47] Source Session 22: Review of Topics | Exam 1 | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Review 1 Figure 2: Review 2 Figure 3: Review 3 Figure 4: Review 4
Captured On [2020-02-05 Wed 22:39] Source Session 24: Functions of Two Variables: Graphs | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Function of 1 variable Figure 2: Example of multivariable functions Figure 3: For simplicity, functions based on 2 or 3 variables Figure 4: Visualize a function of 2 variables...
Captured On [2020-02-05 Wed 22:40] Source Session 25: Level Curves and Contour Plots | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Contour plot definition Figure 2: Level of contour plot Figure 3: Contour plot examples Figure 4: Contour plot guide 2 What is a contour plot? 2.1 Front What is a contour plot?...
Captured On [2020-02-05 Wed 22:40] Source Session 26: Partial Derivatives | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Derivative of a function of 1 variable Figure 2: Approximation formula Figure 3: Partial derivative definition Figure 4: Partial derivative geometrically Figure 5: Example of partial derivative 2 What is a partial function of a multivariable function?...
Captured On [2020-02-06 Thu 10:31] Source Session 27: Approximation Formula | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review partial derivatives Figure 2: Approximation formula Figure 3: Justify approximation formula Figure 4: Tangent line to the same point Figure 5: Tagent plane through 2 tangent lines 2 What is a tangent plane?...
Captured On [2020-02-06 Thu 10:32] Source Session 28: Optimization Problems | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Applications of Partial Derivatives Figure 2: At local min or max Figure 3: Definition of critical point Figure 4: Solve system of equations of partial derivatives Figure 5: What kind of critical point is it?...
Captured On [2020-02-06 Thu 10:32] Source Session 29: Least Squares | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Least squares interpolation Figure 2: Find best “a” and “b” Figure 3: Partial derivatives “a” and “b” Figure 4: Simply minimization of partial derivatives Figure 5: 2x2 Linear system Figure 6: Least square for exponential data...
Captured On [2020-02-05 Wed 19:43] Source Session 3: Uses of the Dot Product: Lengths and Angles | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Computing lengths and angles Figure 2: Resolution of computing an angle Figure 3: Meaning of sign of a dot product Figure 4: Detect orthogonality Figure 5: Plane throught \(O\), perpendicular to \(\vec{A}\)
Captured On [2020-02-06 Thu 10:32] Source Session 30: Second Derivative Test | Part A: Functions of Two Variables, Tangent Approximation and Optimization | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall critical points Figure 2: How do we find global min/man? Figure 3: Second derivative test Figure 4: Squared terms Figure 5: Studying cases Figure 6: When \(4ac - b^{2} > 0\)...
Captured On [2020-02-06 Thu 12:28] Source Session 32: Total Differentials and the Chain Rule | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Differentials and implicit differentiation Figure 2: Example of implicit differentiation Figure 3: Total differential Figure 4: What can do with differentials Figure 5: What can do with differentials (2) Figure 6: Chain Rule...
Captured On [2020-02-06 Thu 12:28] Source Session 34: The Chain Rule with More Variables | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Chain rule with more variables Figure 2: Expansion of chain rule Figure 3: Partial derivative with more variables Figure 4: Example with polar coordinates 2 How is the chain rule for partial derivative with more than one independent variable?...
Captured On [2020-02-06 Thu 12:28] Source Session 35: Gradient: Definition, Perpendicular to Level Curves | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall chain rule Figure 2: Definition of gradient Figure 3: Gradient is perpendicular to level surface Figure 4: Example of Gradient (1) Figure 5: Example of Gradient (2) 2 What is the definition of gradient?...
Captured On [2020-02-06 Thu 12:29] Source Session 36: Proof | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Curve that stays on the level \(w=c\) Figure 2: Chain rule Figure 3: Gradient perpendicular to tangent vector Figure 4: Any vector of tangent plane is perpendicular to gradient 2 How can we proof that gradient is perpendicular to level curves and surfaces?...
Captured On [2020-02-06 Thu 12:29] Source Session 37: Example | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Find tangent plane to surface Figure 2: Gradient is the normal vector of tangent plane Figure 3: Another way throught linear approximation Figure 4: Meaning of approximation 2 How can we proof that gradient is perpendicular to level curves and surfaces?...
Captured On [2020-02-06 Thu 12:29] Source Session 38: Directional Derivatives | Part B: Chain Rule, Gradient and Directional Derivatives | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Directional derivatives Figure 2: Straight line trayectory Figure 3: Directional derivative in direction of \(\vu{u}\) Figure 4: Slope of a slice Figure 5: Directional derivative as a components of gradient Figure 6: Geometrically Figure 7: Fastest increase of \(w\)...
Captured On [2020-02-06 Thu 12:56] Source Session 39: Statement of Lagrange Multipliers and Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Lagrange multipliers Figure 2: Example of using lagrange multipliers Figure 3: Subject to the constraint Figure 4: Both gradient vectors are parallel at level curves Figure 5: System of equations Figure 6: 2 gradiant vector and constraint...
Captured On [2020-02-05 Wed 19:46] Source Session 4: Vector Components | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Review of dot product Figure 2: Components of \(\vec{A}\) along direction \(\vec{u}\) Figure 3: Pendulum problem with projections 2 Which is it the angle between 2 vectors? 2.1 Front Which is it the angle between 2 vectors?...
Captured On [2020-02-06 Thu 12:57] Source Session 40: Proof of Lagrange Multipliers | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Why this method is valid? Figure 2: Tangent to contrained level curve Figure 3: Both gradient vector are parallel Figure 4: Warning: Can’t use second derivatives Figure 5: Compare values at \(f\)