241w-f13_16.1-16.5 (1)

241w-f13_16.1-16.5 (1) - 16.1 Vector fields A vector field...

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1 16.1 Vector fields A vector field in space is a vector-valued 3-variable function ( ) ⟨ ( ) ( ) ( )⟩ Similarly, a vector field in the plane is a vector-valued 2-varible function ( ) ⟨ ( ) ( )⟩ . Examples 1. ( ) 2. Gravitational force: ( ) | |
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2 3. Electric force (Coulomb’s law) ( ) | | Or ( ) | | (electric field of Q) 4. For a function of 2 or 3 variables, is the gradient vector field of .
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3 Definition A vector field F is a conservative vector field if there is a scalar function such that . ( is called the potential function of F.) Example. Let ( ) . Then ( ) | | x . (So a gravitational field is conservative.) We will address the questions (1) how to decide whether a given vector field is conservative; (2) if it is known that a vector field is conservative, how to find its potential function, in the next few sections.
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4 16.2 Line integrals Consider a curve C: ( ) ⟨ ( ) ( ) ( )⟩ . The line integral of f over C is ( ) ( ) . Note: (1) length(C). (2) ( ) ( ( ) ( ) ( )) ( ) ( ) ( ) dt ( ( ))| ( )|
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5 Examples (1) ( ) (2) Let C be graph of . Find .
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