Irrotational Fields were defined [Transparancy 1] as fields which have a curl of zero everywhere. If F is a conservative field, it will automatically be irrotational because the curl of the grad of any scalar field is zero.
Is the converse true? That is, if a field is irrotational, is it necessarily conservative? It turns out that the converse is only partially true, as given by a theorem which holds only for simply connected domains.
This theorem was proved in the lecture [Transparancies 2 and 3] for the case in which a vector field is defined throughout all space. The proof involves assuming the vector field has a curl of zero, defining a scalar field, and showing that the vector field is the gradient of this scalar field.
An example was considered [Transparancy 4], in which a vector field is irrotational (the curl of F is zero), and yet the field is not conservative. The reason is because the domain is not simply-connected and the theorem applies only to simply-connected domains.
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