calculus

Session 67: Proof of Green's Theorem

Captured On [2020-02-06 Thu 16:31] Source Session 67: Proof of Green’s Theorem | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Proof of Green’s Theorem Figure 2: 2 observations Figure 3: Proof of one part Figure 4: Cut (R) “vertically simple” Figure 5: Main step of prove Figure 6: Line integrals...

Session 68: Planimeter: Green's Theorem and Area

Captured On [2020-02-06 Thu 16:32] Source Session 68: Planimeter: Green’s Theorem and Area | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Planimeter 2 What is a planimeter? 2.1 Front What is a planimeter? Define it, and write its equation 2.2 Back It’s an instruments for measuring areas through closing perimeters. This instruments uses the Green’s Theorem, and the equation...

Session 69: Flux in 2D

Captured On [2020-02-06 Thu 16:33] Source Session 69: Flux in 2D | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Flux presentation Figure 2: $\vu{n}$ as clockwise from $\vu{T}$ Figure 3: Flux and Work comparison Figure 4: Intepretation of Flux Figure 5: Visualization of Flux Figure 6: What flows across $C$...

Session 7: Cross Products

Captured On [2020-02-05 Wed 19:50] Source Session 7: Cross Products | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Cross product of 2 vectors in 3-space Figure 2: Area of parallelogram with cross product Figure 3: Right hand method for direction of cross product Figure 4: Another look at volume Figure 5: Volume of parallelopiped...

Session 70: Normal Form of Green's Theorem

Captured On [2020-02-06 Thu 16:34] Source Session 70: Normal Form of Green’s Theorem | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: What is the Green’s Theorem for flux Figure 2: Green’s Theorem for flux Figure 3: Green’s Theorem in normal form vs tangential form Figure 4: Proof of Green’s Theorem for flux...

Session 71: Extended Green's Theorem: Boundaries with Multiple Pieces

Captured On [2020-02-06 Thu 16:35] Source Session 71: Extended Green’s Theorem: Boundaries with Multiple Pieces | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: More about validity of Green’s Theorem Figure 2: $\vb{F}$ defined everywhere in $R$ Figure 3: Use of Green’s Theorem Figure 4: Remove the region where $\vb{F}$ is not defined...

Session 72: Simply Connected Regions and Conservative Fields

Captured On [2020-02-06 Thu 16:36] Source Session 72: Simply Connected Regions and Conservative Fields | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Connected region in the plane Figure 2: Where can I apply Green’s Theorem? Figure 3: Correct definition of $\vb{F}$ conservative 2 What is a simply-connected region? 2.1 Front What is a simply-connected region?...

Session 73: Review Double Integrals and Line Integrals in the Plane

Captured On [2020-02-06 Thu 18:03] Source Session 73: Exam Review | Exam 3 | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare Figure 1: Review 1 Figure 2: Review 2 Figure 3: Review 3 Figure 4: Review 4 Figure 5: Review 5 Figure 6: Review 6 Figure 7: Review 7 Figure 8: Review 8 Figure 9: Review 9 Figure 10: Review 10...

Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates

Captured On [2020-02-06 Thu 16:38] Source Session 74: Triple Integrals: Rectangular and Cylindrical Coordinates | Part A: Triple Integrals | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Triple integrals Figure 2: Example of triple integral Figure 3: Setting up interated integral for triple integral Figure 4: Find shadow in $xy\text{-plane}$ Figure 5: Computing iterated integral Figure 6: Better use polar coordinates for triple coordinates...

Session 75: Applications and Examples

Captured On [2020-02-06 Thu 17:02] Source Session 75: Applications and Examples | Part A: Triple Integrals | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Mass on triple integral Figure 2: Average value on triple integral Figure 3: Center of mass on triple integral Figure 4: Moment of inertia on triple integral Figure 5: Moment of inertia about axis...

Session 76: Spherical Coordinates

Captured On [2020-02-06 Thu 17:03] Source Session 76: Spherical Coordinates | Part A: Triple Integrals | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Spherical coordinates Figure 2: On a sphere $\rho = a$ Figure 3: Transform from cylindrical to spherical coordinates Figure 4: Transform from spherical to rectangular coordinates Figure 5: Examples of figures on spherical coordinates...

Session 77: Triple Integrals in Spherical Coordinates

Captured On [2020-02-06 Thu 17:04] Source Session 77: Triple Integrals in Spherical Coordinates | Part A: Triple Integrals | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Triple integral in spherical coordinates Figure 2: Surface area on sphere of radius $a$ Figure 3: Surface element Figure 4: Volume element Figure 5: Example of volume of unit sphere above $z = 1 / \sqrt{2}$...

Session 78: Applications: Gravitational Attraction

Captured On [2020-02-06 Thu 17:05] Source Session 78: Applications: Gravitational Attraction | Part A: Triple Integrals | 4. Triple Integrals and Surface Integrals in 3-Space | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Graviational force exerted by $\Delta M$ Figure 2: Gravitational force vector Figure 3: Setting up triple integral on an axis of symmetry Figure 4: Use spherical coordinates Figure 5: Newton’s Theorem Figure 6: Newston’s Theorem (II)

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 8: Equations of Planes

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?...

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