LineIntegral

# LineIntegral - Line Integral. Ian R. Gatland, Georgia...

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Unformatted text preview: Line Integral. Ian R. Gatland, Georgia Institute of Technology Physics 3122 notes, September 2, 2010 Consider a line integral, ∫ v ⋅ d , where the points on the line are specified in terms of some parameter, t, as x ( t ), y ( t ), and z( t ) . Then ˆ ˆ ˆ ∫ v ⋅ d = ∫ v ( x, y, z)⋅ ( xdx + ydy + zdz) € ȹ dx dy dz ȹ ˆ ˆ ˆ = ∫ v ( x ( t ), y ( t ), z( t ))⋅ ȹ x dt + y dt + z dt ȹ € ȹ dt dt dt Ⱥ ȹ dx dy dz ȹ ˆ ˆ ˆ = ∫ v ( x ( t ), y ( t ), z( t ))⋅ ȹ x +y + z ȹ dt ȹ dt dt dt Ⱥ where the derivatives are expressed in terms of t. € In many cases the parameter may be one of the coordinates so if t is x then ˆˆ ˆ ∫ v ⋅ d = ∫ v ( x, y ( x ), z( x ))⋅ ȹ x + y dx + z dx ȹdx ȹ Ⱥ ȹ where the derivatives are expressed in terms of x. € There are similar results when t is y and when t is z. dy dz ȹ ...
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