3B: Vector Fields and Line Integrals

A vector field attaches a vector to each point. For example, the sun has a gravitational field, which gives its gravitational attraction at each point in space. The field does work as it moves a mass along a curve. We will learn to express this work as a line integral and to compute its value.

In physics, some force fields conserve energy. Such conservative fields are determined by their potential energy functions. We will define what a conservative field is mathematically and learn to identify them and find their potential function.

Last Modification: 2020-09-17 Thu 12:25

Captured On: [2020-01-18 Sat]
Source: Part B: Vector Fields and Line Integrals | 3. Double Integrals and Line Integrals in the Plane | Multivariable Calculus | Mathematics | MIT OpenCourseWare

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 60: Fundamental Theorem for Line Integrals

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...

Session 61: Conservative Fields, Path Independence, Exact Differentials

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?...

Session 62: Gradient Fields

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...

Session 63: Potential Functions

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...

Session 64: Curl

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...