calculus

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?...

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?...