maths

Captured On [2020-02-06 Thu 13:32] Source Session 60: Fundamental Theorem for Line Integrals | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Special case where vector field is gradient field Figure 2: Fundamental Theorem of Calculus for Line Integrals Figure 3: Proof of FTC for Line Integrals Figure 4: Proof of FTC for Line Integrals - 2...

Captured On [2020-02-06 Thu 13:33] Source Session 61: Conservative Fields, Path Independence, Exact Differentials | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Gradient fields in physics Figure 2: Meaning of conservativeness Figure 3: Equivalent properties I Figure 4: Equivalent properties II Figure 5: Equivalent properties III 2 How can we proof that path independence is equivalent to conservative?...

Captured On [2020-02-06 Thu 13:33] Source Session 62: Gradient Fields | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall FCT for line integral with gradient fields Figure 2: Meaning of conservative field Figure 3: How to know if vector fields is a gradient field Figure 4: Test for gradient field...

Captured On [2020-02-06 Thu 13:34] Source Session 63: Potential Functions | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Finding the potential Figure 2: Choosing a path Figure 3: Choosing a easiest path Figure 4: Path \(C_{1}\) Figure 5: Path \(C_{2}\) Figure 6: The potential function Figure 7: Using antiderivatives...

Captured On [2020-02-06 Thu 13:34] Source Session 64: Curl | Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall \(\vb{F}\) using curl Figure 2: Definition of curl Figure 3: Curl for velocity field Figure 4: Examples of curl Figure 5: What measure the curl Figure 6: Physics derivative measures...

Captured On [2020-02-06 Thu 13:53] Source Session 65: 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: Line integral for closed path Figure 2: Green’s Theorem Figure 3: Green’s Theorem (II) Figure 4: Warning and example Figure 5: Computing directly Figure 6: Using Green’s Theorem Figure 7: Solving with geometry and simmetry...

Captured On [2020-02-06 Thu 16:29] Source Session 66: Curl(F) = 0 Implies Conservative | Part C: Green’s Theorem | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Special case of Green Theorem Figure 2: Proof of \(\vb{F}\) conservative when curl \(\vb{F}\) is \(0\) Figure 3: Consequence of \(\text{curl} \vb{F} = 0\) Figure 4: Cannot Green theorem when \(\vb{F}\) is not defined at point inside the region...

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...

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...

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 \(C\)...

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...

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...

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...

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?...

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...

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...

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...

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...

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 \(z = 1 / \sqrt{2}\)...

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)