1 / 12

Chapter 16 – Vector Calculus

Chapter 16 – Vector Calculus. 16.4 Green’s Theorem. Objectives: Understand Green’s Theorem for various regions Apply Green’s Theorem in evaluating a line integral. Introduction.

daisy
Download Presentation

Chapter 16 – Vector Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 16 – Vector Calculus 16.4 Green’s Theorem • Objectives: • Understand Green’s Theorem for various regions • Apply Green’s Theorem in evaluating a line integral 16.4 Green’s Theorem

  2. Introduction • George Green worked in his father’s bakery from the age of 9 and taught himself mathematics from library books. • In 1828 published privately what we know today as Green’s Theorem. It was not widely known at the time. • At 40, Green went to Cambridge and died 4 years after graduating. • In 1846, Green’s Theorem was published again. • Green was the first person to try to formulate a mathematical theory of electricity and magnetism. George Green (1793 – 1841) 16.4 Green’s Theorem

  3. Introduction • Green’s Theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. • We assume that Dconsists of all points inside C as well as all points on C. 16.4 Green’s Theorem

  4. Introduction • In stating Green’s Theorem, we use the convention: • The positive orientation of a simple closed curve C refers to a single counterclockwise traversal of C. • Thus, if C is given by the vector function r(t), a ≤t ≤b, then the region D is always on the left as the point r(t) traverses C. 16.4 Green’s Theorem

  5. Green’s Theorem • Let C be a positively oriented, piecewise-smooth, simple closed curve in the plane and let D be the region bounded by C. • If P and Q have continuous partial derivatives on an open region that contains D, then 16.4 Green’s Theorem

  6. Green’s Theorem Notes • The notation is sometimes used to indicate that the line integral is calculated using the positive orientation of the closed curve C. 16.4 Green’s Theorem

  7. Green’s Theorem Notes • Another notation for the positively oriented boundary curve of D is∂D. • So,the equation in Green’s Theorem can be written as equation 1: 16.4 Green’s Theorem

  8. Green’s Theorem and FTC • Green’s Theorem should be regarded as the counterpart of the Fundamental Theorem of Calculus (FTC) for double integrals. • Compare Equation 1 with the statement of the FTC Part 2 (FTC2), in this equation: • In both cases, • There is an integral involving derivatives (F’,∂Q/∂x, and ∂P/∂y) on the left side. • The right side involves the values of the original functions (F, Q, and P) only on the boundary of the domain. 16.4 Green’s Theorem

  9. Area • Green’s Theorem gives the following formulas for the area of D: 16.4 Green’s Theorem

  10. Example 1 – pg. 1060 #6 • Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. 16.4 Green’s Theorem

  11. Example 2 – pg. 1060 #8 • Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. 16.4 Green’s Theorem

  12. Example 3 – pg. 1060 #12 • Use Green’s Theorem to evaluate . • Check the orientation of the curve before applying the theorem. 16.4 Green’s Theorem

More Related