14.1 Notes - Calculus III Section 14.1 Vector Field Let M...

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Calculus III – Section 14.1 Vector Field Let M and N be functions of two variables x and y, defined on a plane region R. The function F defined by F (x,y) = M i + N j Is called a vector field over R . As usual, we can extend the definition of a vector field to 3-space. A vector field is continuous at a point if each of its component functions is continuous at that point. You have already seen at least one example of a vector field, namely f . Example: Sketch several representative vectors in the vector field: j y i x y x F + = ) , ( . Conservative Vector Fields A vector field F is called conservative if there exists a differentiable function f such that f F = . The function f is called the potential function for F . Example: ( 29 ( 29 j xy x y i y xy y x F 2 2 3 3 3 4 6 ) , ( - + + - = is a conservative vector field. Verify that 3 2 2 3 2 ) , ( xy y x y y x f - + = is a potential function for F . Test for Conservative Vector Field in the Plane Let M and N have continuous first partial derivatives on an open disk R. The vector field given by
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14.1 Notes - Calculus III Section 14.1 Vector Field Let M...

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