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15a). This procedure can be extended to calculate the circulation around a general curve � in the (�, �)-plane. This result for the general curve in the (�, �)-plane is given by ( ) �� �� (� �� + � ��) = − �� ��. 23) is Green’s theorem, relating a line integral to the corresponding area integral. The transformation from a line integral to a surface integral in three-dimensional space is governed by Stokes’ theorem ∮ �⃗ . 24) where �̂ � is the area vector normal to the surface and is positive when it is pointing outward from the enclosed volume and equal in magnitude to the surface area.

24) is not valid if contains regions where �⃗ or its derivatives are infinite. e. (∇ × �⃗ ) is zero at all points in the region bounded by �, then the circulation is zero. 25) Basic Equations of Motion 37 and the flow contains no singularities, then the flow is said to be irrotational. Thus for irrotational flows, the line integral ∮ �⃗ . �⃗� is independent of path. The line integral depends only on its end point. e. �⃗ = ∇�. Extension of this to three-dimensional flows is trivial and is left as an exercise to the reader.

Note: (i) The variation of � along �� with � can also be considered, but the contributions mutually cancel out, leaving the above. 15a). This procedure can be extended to calculate the circulation around a general curve � in the (�, �)-plane. This result for the general curve in the (�, �)-plane is given by ( ) �� �� (� �� + � ��) = − �� ��. 23) is Green’s theorem, relating a line integral to the corresponding area integral. The transformation from a line integral to a surface integral in three-dimensional space is governed by Stokes’ theorem ∮ �⃗ .