calculus

Captured On [2020-02-06 Thu 17:17] Source Session 85: Physical Meaning of Flux; Del Notation | Part B: Flux and the Divergence Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall divergence theorem Figure 2: “Del” notation Figure 3: Physical interpretation of divergence Figure 4: Amount of fluid leaving \(D\) per unit time 2 What means “source rate”?...

Captured On [2020-02-06 Thu 17:18] Source Session 86: Proof of the Divergence Theorem | Part B: Flux and the Divergence Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Proof of divergence theorem Figure 2: \(D\) is vertically simple Figure 3: Checking both sides has same outcome Figure 4: Flux at top side Figure 5: Flux at bottom side...

Captured On [2020-02-06 Thu 17:19] Source Session 87: Diffusion Equation | Part B: Flux and the Divergence Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Introduction to Diffusion Equation Figure 2: Heat equation Figure 3: How relate \(\vb{F}\) with \(\pdv{u}{t}\) Figure 4: Using divergence theorem Figure 5: Setting equations for divergence equality Figure 6: Divergence of flow...

Captured On [2020-02-06 Thu 17:22] Source Session 88: Line Integrals in Space | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Line Integrals in Space Figure 2: Evaluation of Line Integrals Figure 3: Example of Line Integrals Figure 4: Example, same \(\vb{F}\) changing path Figure 5: Evaluation of example 2...

Captured On [2020-02-06 Thu 17:39] Source Session 89: Gradient Fields and Potential Functions | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Test for Gradient Fields Figure 2: Criterion for Gradient Field Figure 3: Example for criterion gradient field Figure 4: Find the Potential Figure 5: FTC and Path method...

Captured On [2020-02-05 Wed 21:13] Source .Session 9: Matrix Multiplication | Part B: Matrices and Systems of Equations | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What is a matrix? Figure 2: Change coordinate systems Figure 3: Entries in matrix multiplication Figure 4: Nmemotecnic rule for matrix multiplication 2 How we can set up 2 matrix for its multiplication? 2.1 Front How we can set up 2 matrix for its multiplication?...

Captured On [2020-02-06 Thu 17:41] Source Session 90: Curl in 3D | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Curl in 3D Figure 2: \(\vb{F}\) in simply-connected region Figure 3: Del notation Figure 4: Mnemotecnic formula for curl \(\vb{F}\) Figure 5: Geometric meaning of curl 2 Is the \(\text{curl}\vb{F}\) a scalar function?...

Captured On [2020-02-06 Thu 17:42] Source Session 91: Stokes’ Theorem | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall \(\text{curl}\vb{F}\) Figure 2: Curl examples Figure 3: Stokes’ Theorem Figure 4: Orientation of \(S\) and \(C\) Figure 5: Right-hand rule for \(S\) and \(C\) orientation Figure 6: Comparing Stokes with Green...

Captured On [2020-02-06 Thu 17:43] Source Session 92: Proof of Stokes’ Theorem | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Why Stokes is true? Figure 2: Strategy for proofing Stokes’ Theorem 2 Why Stokes is true? 2.1 Front Why Stokes is true? 2.2 Back Know it for \(C\), \(S\) in \(xy\text{-plane}\) (Green’s Theorem) also for \(C\), \(S\) in any plane using that work, flux, curl make sense independently of the coordinate system Strategy of proofing Given any \(S\): decompose it into tiny, almost flat pieces Sum of work around each piece = work along \(C\) Similar to Green’s Theorem Sum of flux through each piece = flux through \(S\)

Captured On [2020-02-06 Thu 17:43] Source Session 93: Example | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Example of Stokes’ Theorem Figure 2: Computing the surface integral Figure 3: Same result with Stokes’ Theorem

Captured On [2020-02-06 Thu 17:44] Source Session 94: Simply Connected Regions; Topology | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Stokes and Path Independence Figure 2: Examples of simply-connected spaces Figure 3: Theorem for path independence of line integrals Figure 4: Proof of path independence of line integrals of conservative \(\vb{F}\)...

Captured On [2020-02-06 Thu 17:45] Source Session 95: Stokes’ Theorem and Surface Independence | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Stokes and surface independence Figure 2: Check with divergence theorem Figure 3: Divergence of curl of vector fiels is 0 Figure 4: For real vectors 2 Why for Stokes’ Theorem we can use any surface (surface independence)?...

Captured On [2020-02-06 Thu 18:05] Source Session 96: Summary of Multiple Integration | Part C: Line Integrals and Stokes’ Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Different multiple integrals Figure 2: Applications of multiple integrals Figure 3: Different surfaces Figure 4: Getting \(\vu{n}\dd{S}\) Figure 5: Divergence and Stokes’ Theorem Figure 6: FTC for line integrals

Captured On [2020-02-06 Thu 18:16] Source Session 97: Curl and Physics | Physics Applications | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Recall what measures curl of velocity field Figure 2: Curl of velocity field Figure 3: Curl on force field Figure 4: Using curl of translation force to produce rotation Figure 5: Consecuences if force field is derive from a potential

Captured On [2020-02-06 Thu 18:18] Source Session 98: Maxwell’s Equations | Physics Applications | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Maxwell’s Equations Figure 2: Gauss-Coulumb Law Figure 3: Faraday’s Law Figure 4: Remark