This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Lecture XIX Visualizing Vector Fields; Line Integrals Visualizing Vector Fields Recall that a vector field in E 2 is a function of the form F ( x, y ) = f 1 ( x, y ) ˆ i + f 2 ( x, y ) ˆ j. We define two concepts that help us visualize vector fields. Definition 1 Let C be a directed smooth curve in E 2 , and let F be a vector field in E 2 . Then C is called an integral curve of F if, at any point P on C , F ( P ) = 0 and F ( P ) has the same direction F ( P ) is tangent to the curve C , i.e. as T ˆ P , the unit tangent vector to C at P . Recall that C is the class of all continuous scalar functions and that C 1 is the class of all differentiable functions with continuous partial derivatives. We say that the vector field F as defined above is in C 1 if f 1 , f 2 are in C 1 . Lemma 1 For any point P such that F ( P ) = 0 , there exists an integral curve for F through P . F ˆ Let us take = x ˆ i + yj = rr ˆ. The integral curves of F are rays comming out of 1 the origin. F passes the derivative test, and it is easy to see that...
View
Full Document
 Two '04
 Duorg
 Math, Derivative, Integrals, Vector Space, Gradient, integral curve

Click to edit the document details