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Session 41: Advanced Example

Captured On [2020-02-06 Thu 12:57] Source Session 41: Advanced Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Build pyramid Figure 2: First draft, can’t solve Figure 3: Another approximation to the solution Figure 4: Function to minimize and constrain Figure 5: Solution

Session 42: Constrained Differentials

Captured On [2020-02-06 Thu 12:57] Source Session 42: Constrained Differentials | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Non-independent variables Figure 2: $z$ as a function of $x$ and $y$ Figure 3: Take differential Figure 4: Partial derivatives Figure 5: In general 2 What problem can we find when we are computing partial derivatives with non-independent variables?...

Session 43: Clearer Notation

Captured On [2020-02-06 Thu 12:59] Source Session 43: Clearer Notation | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Introduction to confussion Figure 2: Equals variable but different partial derivatives Figure 3: Need clearer notation 2 What differences are there between formal and actual partial derivatives? 2.1 Front What differences are there between formal and actual partial derivatives?...

Session 44: Example

Captured On [2020-02-06 Thu 12:59] Source Session 44: Example | Part C: Lagrange Multipliers and Constrained Differentials | 2. Partial Derivatives | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Area of triangle Figure 2: Rate of change of area wrt $\theta$ Figure 3: Keeping the right triangle Figure 4: Keeping the right triangle (2) Figure 5: Method 0: Substituting values Figure 6: Systematic methods: Differentials (1)...

Session 45: Review of Partial Derivatives

Captured On [2020-02-06 Thu 18:02] Source Session 45: Review of Topics | Exam 2 | 2. Partial Derivatives | 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 Figure 11: Review 11...

Session 47: Definition of Double Integration

Captured On [2020-02-06 Thu 13:14] Source Session 47: Definition of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Recall integrals of 1 variable function Figure 2: Meaning of double integral Figure 3: Small pieces of area Figure 4: Computing double integrals taking slices Figure 5: Iterated integrals 2 Can we compute a double integrals as a limit of Riemann sums?...

Session 48: Examples of Double Integration

Captured On [2020-02-06 Thu 13:14] Source Session 48: Examples of Double Integration | Part A: Double Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare 1 Chalkboard Figure 1: Example of iterated integral Figure 2: Inner integral Figure 3: Outer integral Figure 4: Same function, other region Figure 5: Inner function, more complex region Figure 6: Inner cont., outer function...

Session 49: Exchanging the Order of Integration

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?...

Session 5: Area and Determinants in 2D

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....

Session 50: Double Integrals in Polar Coordinates

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 $dr$...

Session 51: Applications: Mass and Average Value

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 $\delta$...

Session 52: Applications: Moment of Inertia

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 $x\text{-axis}$...

Session 53: Change of Variables

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...

Session 54: Example: Polar Coordinates

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...

Session 55: Example

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...

Session 56: Vector Fields

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?...

Session 57: Work and Line Integrals

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...

Session 58: Geometric Approach

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...

Session 59: Example: Line Integrals for Work

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...

Session 6: Volumes and Determinants in Space

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 $\vb{a} \cross \vb{b} = 0$?...