calculus

Captured On [2020-02-06 Thu 13:14] Source Session 49: Exchanging the Order of Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Exchanging order of integration Figure 2: Example of exchanging order of integration Figure 3: Example conclusion 2 How can we exchanging the order of integration? 2.1 Front How can we exchanging the order of integration?...

Captured On [2020-02-05 Wed 19:47] Source Session 5: Area and Determinants in 2D | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area within 2 vectors Figure 2: Rotated vector \(\vec{A}\) Figure 3: Area as determinant of 2 vectors Figure 4: Area of parallelogram and triangle 2 What is the area between 2 vectors? 2....

Captured On [2020-02-06 Thu 13:15] Source Session 50: Double Integrals in Polar Coordinates | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Double integral as interated integral \(\dd{y}\dd{x}\) Figure 2: Function in polar coordinates Figure 3: Setting up interated integrals on polar coordinates (1) Figure 4: Relationship between \(dA\) and \(d\theta\) and \(dr\)...

Captured On [2020-02-06 Thu 13:15] Source Session 51: Applications: Mass and Average Value | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area of a region \(R\) Figure 2: Mass of a flat object with density \(\delta\) Figure 3: Average value of \(f\) Figure 4: Center of mass of a (planar) object with density \(\delta\)...

Captured On [2020-02-06 Thu 13:15] Source Session 52: Applications: Moment of Inertia | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Moment of inertia Figure 2: Kinetic energy Figure 3: Moment of inertia for each \(\Delta A\) Figure 4: Moment of inertia about the origin Figure 5: Moment of inertia about \(x\text{-axis}\)...

Captured On [2020-02-06 Thu 13:16] Source Session 53: Change of Variables | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Change variable in double integral Figure 2: Basic example Figure 3: Basic example continuation Figure 4: Find scaling factor Figure 5: Linear change of variables Figure 6: Example of linear change of variable...

Captured On [2020-02-06 Thu 13:16] Source Session 54: Example: Polar Coordinates | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Polar coordinates Figure 2: Scaling factor Figure 3: Recall reciprocal rule for partial derivatives 2 Proof \(dA\) in polar coordinates with the Jacobian 2.1 Front Proof $\dd{A}$ in polar coordinates with the Jacobian...

Captured On [2020-02-06 Thu 13:16] Source Session 55: Example | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area element Figure 2: Integrand in terms of \(u,v\) Figure 3: Bounds Figure 4: Bounds switching to \(uv\) picture 2 How can we compute \(\Delta A\) in another \(u,v\) coordinates 2.1 Front How can we compute $\Delta A$ in another $u,v$ coordinates...

Captured On [2020-02-06 Thu 13:29] Source Session 56: Vector 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: Vector fields meaning Figure 2: Examples of vector fields (1) Figure 3: Examples of vector fields (2) Figure 4: Examples of vector fields (4) 2 How are defined a vector field?...

Captured On [2020-02-06 Thu 13:30] Source Session 57: Work and 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: Work and line integrals Figure 2: Work as integral Figure 3: Example of work Figure 4: Example cont Figure 5: Another way of calculate line integrals Figure 6: Compute line integrals in terms of one variable...

Captured On [2020-02-06 Thu 13:31] Source Session 58: Geometric Approach | 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: Geometric approach Figure 2: Example \(C\) circle of radius \(a\), and force radially away Figure 3: Same \(c\), but force are tanget to the circle Figure 4: Computing is more long that geometric approach in this case...

Captured On [2020-02-06 Thu 13:32] Source Session 59: Example: Line Integrals for Work | 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: Line integrals for work Figure 2: Example of line integrals for work Figure 3: First trajectory Figure 4: Second trajectory Figure 5: Second trajectory cont Figure 6: Third trajectory...

Captured On [2020-02-05 Wed 19:49] Source Session 6: Volumes and Determinants in Space | Part A: Vectors, Determinants and Planes | 1. Vectors and Matrices | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Determinant in space Figure 2: Volume of parallelepiped 2 What is the volume between 3 vectors? 2.1 Front What is the volume between 3 vectors? 2.2 Back With determinants \(\abs{\det(\vb{A}, \vb{B}, \vb{C})}\) With cross product Volume = area(base) height \(V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \hat{n})\) \({\displaystyle \hat{n} = \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}}\) \({\displaystyle V = \abs{\vb{A} \cross \vb{B}} (\vb{C} \cdot \frac{\vb{A} \cross \vb{B}}{\abs{\vb{A} \cross \vb{B}}}) = \vb{C} \cdot (\vb{A} \cross \vb{B})}\) 3 What happens if \(\vb{a} \cross \vb{b} = 0\)?...

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...