| The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
|
Home  Vector Calculus Independence of Path Theorem 1 |
|
 Search the VIAS Library | Index
|
  |
Theorem 1
THEOREM 1 A vector field Pi + Qj has a potential function if and only if We have already proved one direction. We postpone the proof of the other direction until later. By definition, grad f = Pi + Qj if and only if df = P dx + Q dy. In general, an expression P dx + Q dy is called a differential form. A differential form is called an exact differential if it is equal to the total differential df of some function f(x, y). Using this terminology, Theorem 1 states that: P dx + Q dy is an exact differential if and only if ∂P/∂y = ∂Q/∂x.
|
|
| Home Vector Calculus Independence of Path Theorem 1 |
|
Last Update: 2006-11-22