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Chapter 16 – Vector Calculus

Chapter 16 – Vector Calculus. 16.2 Line Integrals. Objectives: Understand various aspects of line integrals in planes, space, and vector fields. Line Integrals. They were invented in the early 19th century to solve problems involving: Fluid flow Forces Electricity Magnetism.

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Chapter 16 – Vector Calculus

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  1. Chapter 16 – Vector Calculus 16.2 Line Integrals • Objectives: • Understand various aspects of line integrals in planes, space, and vector fields 16.2 Line Integrals

  2. Line Integrals • They were invented in the early 19th century to solve problems involving: • Fluid flow • Forces • Electricity • Magnetism 16.2 Line Integrals

  3. Line Integrals • We start with a plane curve C given by the parametric equations (Equation 1) x = x(t) y = y(t) a ≤ t ≤ b • Equivalently, C can be given by the vector equation r(t) = x(t) i + y(t) j. • We assume that Cis a smooth curve. • This means that r′is continuous and r′(t) ≠ 0. 16.2 Line Integrals

  4. Definition • If f is defined on a smooth curve C given by Equations 1, the line integral of falong Cis: if this limit exists. Then, this formula can be used to evaluate the line integral. 16.2 Line Integrals

  5. Example 1 – pg. 1096 #2 • Evaluate the line integral, where C is the given curve. 16.2 Line Integrals

  6. Line Integrals in Space • We now suppose that C is a smooth space curve given by the parametric equations x = x(t) y = y(t) a ≤ t ≤ b or by a vector equation r(t) = x(t) i + y(t) j + z(t) k 16.2 Line Integrals

  7. Line Integrals in Space • Suppose f is a function of three variables that is continuous on some region containing C. • Then,we define the line integral of f along C(with respect to arc length) in a manner similar to that for plane curves: • We evaluate it using 16.2 Line Integrals

  8. Example 2 – pg. 1096 #10 • Evaluate the line integral, where C is the given curve. 16.2 Line Integrals

  9. Example 3 • Evaluate the line integral, where Cis the given curve. 16.2 Line Integrals

  10. Line Integrals of Vector Fields • Definition - Let F be a continuous vector field defined on a smooth curve Cgiven by a vector function r(t), a ≤ t ≤ b. • Then, the line integral of F along Cis: 16.2 Line Integrals

  11. Notes • When using Definition 13 on the previous slide, remember F(r(t)) is just an abbreviation for F(x(t), y(t), z(t)) • So, we evaluate F(r(t)) simply by putting x = x(t), y = y(t), and z = z(t) in the expression for F(x, y, z). • Notice also that we can formally write dr = r′(t) dt. 16.2 Line Integrals

  12. Example 4 – pg. 1097 #20 • Evaluate the line integral , where C is the given by the vector function r(t). 16.2 Line Integrals

  13. Example 5 – pg. 1097 #22 • Evaluate the line integral , where C is the given by the vector function r(t). 16.2 Line Integrals

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