Review - Path Independence

Review - Path Independence - Vector Calculus: Path...

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Vector Calculus: Path Independence The line integral of the scalar field ( 29 f x,y along a curve ( 29 C x,y may be evaluated directly if both functions can be parametrized easily: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 , , , , ' ' ' C C f x y z ds f x t y t z t x t y t z t dt = + + [1] For a vector field ( 29 F x,y,z v , a scalar line integral can also be evaluated, depending on the condition imposed on the direction of F v . If F v is required to be tangent to the curve ( 29 C x,y at all times, then an inner product is formed between F v and the unit tangent vector T v : x y z C C C C dr dt F Tds F ds F dr F dx F dy F dz dt ds = = = + + v v v v v v [2] A conservative vector field is a vector field which is the gradient of a scalar field. If ( 29 F x,y,z v is conservative such that ( 29 F f x,y,z = ∇ v v , then equation [2] can be simplified, after parametrizing ( 29 C x,y
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Review - Path Independence - Vector Calculus: Path...

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