4B: Flux and the Divergence Theorem

Here we will extend Green’s theorem in flux form to the divergence (or Gauss’) theorem relating the flux of a vector field through a closed surface to a triple integral over the region it encloses. Before learning this theorem we will have to discuss the surface integrals, flux through a surface and the divergence of a vector field.

Last Modification: 2020-09-17 Thu 12:33

Captured On: [2020-01-18 Sat]
Source: Part B: Flux and the Divergence Theorem | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare

Session 79: Vector Fields in Space

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?...

Session 80: Flux Through a Surface

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...

Session 81: Calculating Flux; Finding ndS

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...

Session 82: ndS for a Surface z = f(x, y)

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...

Session 83: Other Ways to Find ndS

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 $\vu{n} \dd{S}$...

Session 84: Divergence Theorem

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 $\vb{F} = M \vu{i} + N \vu{j} + P \vu{k}$...

Session 85: Physical Meaning of Flux; Del Notation

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”?...

Session 86: Proof of the Divergence Theorem

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...

Session 87: Diffusion Equation

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...