Lecture 2

# Lecture 2 - Integrals Gradient topointb Line LineIntegral b...

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ne Integral Line Integral work Δ W is the line integral from point a point b to point b a b b l = Δ a W dl F

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= c b l l + Δ b a W dl F dl F Cartesian coordinates z y x u u u l d dz dy dx + + y 4 c x 4 0 0 a b
ne Integral Example Line Integral Example y 4 + = Δ c b W dl F dl F c b a = x 0 a b z y = z y x u u u F 0 4 3 + + 4 0 z y x u u u l d dz dy dx + + () 0 , 3 x 1 ( ) 3 , 3 y = Δ 0 , 0 3 dx W 21 = + 0 , 3 4 dy

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Closed line integral work Δ W is the line integral from point a to point b continue on to a 0 = = Δ dl F W conservative field 0 = Δ dl F W Non conservative field
Conservative Field R a I V R b R c

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ds F s ds F z ds y y F x
y u F y F = = aa dxdz F u u ds F ∫ ∫ = y a y y y 00 ∫ ∫ + = y y y y dxdz F 0 u u z 3 Fa = Δ s F y Δ s x cube a 3

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herical Coordinates Spherical Coordinates
v ∫∫∫ dv Q v ρ z y x

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Derivative Operations directional derivative value of the rate of change of scalar quantity in a given direction gradient direction and value of the maximum rate of change of scalar quantity ivergence urce r nk f a vector divergence source or sink of a vector field rl irection and magnitude of tation curl direction and magnitude of

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Lecture 2 - Integrals Gradient topointb Line LineIntegral b...

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