Captured On [2020-02-06 Thu 17:08] Source Session 79: Vector Fields in Space | 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: Vector fields in space Figure 2: Force fields Figure 3: Graviational field equation and more examples Figure 4: Velocity and gradient fields 2 What is a force field in 3-space?...
Captured On [2020-02-05 Wed 19:52] Source Session 8: Equations of Planes | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Equation of plane Figure 2: Another solution equation of plane 2 How can we find the equation of the plane containing 3 points? 2.1 Front How can we find the equation of the plane containing 3 points?...
Captured On [2020-02-06 Thu 17:11] Source Session 80: Flux Through a Surface | 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 flux in 2D Figure 2: Flux of \(\vb{F}\) across \(\Delta S\) Figure 3: Flux of \(\vb{F}\) across the surface Figure 4: Example the flux across the sphere...
Captured On [2020-02-06 Thu 17:12] Source Session 81: Calculating Flux; Finding ndS | 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: Flux on not constant \(\vb{F} \cdot \vu{n}\) Figure 2: \(\dd{S}\) in spherical coordinates Figure 3: Computing flux Figure 4: Finding \(\vu{n}\dd{S}\) - I Figure 5: Finding \(\vu{n}\dd{S}\) - II...
Captured On [2020-02-06 Thu 17:14] Source Session 82: ndS for a Surface z = f(x, y) | 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: Flux of \(\vb{F}\) through surface \(S\) Figure 2: Normal unit vector and surface element Figure 3: Getting \(\vu{n} \dd{S}\) - I Figure 4: Cross product of surface vectors...
Captured On [2020-02-06 Thu 17:15] Source Session 83: Other Ways to Find ndS | 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: Generic description of \(S\) Figure 2: Sides of surface Figure 3: Knowing the normal vector of the surface Figure 4: Surface element Figure 5: Writing \(\vu{n} \dd{S}\)...
Captured On [2020-02-06 Thu 17:16] Source Session 84: 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: Divergence theorem Figure 2: Equation of divergence theorem Figure 3: Example of divergence theorem 2 Which is the divergence of \(\vb{F}\)? 2.1 Front Which is the divergence of $\vb{F}$? Let \(\vb{F} = M \vu{i} + N \vu{j} + P \vu{k}\)...
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