In 3 dimensions, a simply connected domain is one in which any closed loop can be shrunk down to a point without getting hung-up somewhere. Therefore, a domain with a spherical hole in it is actually simply connected. An example of a non simply connected domain is one in which the z-axis is excluded, for example the domain of a magnetic field created by an infinitely long wire positioned along the z-axis. In this case, any loop which goes around the z-axis will become stuck on the z-axis when we try to shrink it to a point.
Conservative vector fields were defined [Transparancy 1] as vector fields which can be expressed as the gradient of a scalar field. (This scalar field is often called a "potential".)
We then looked in detail at an important theorem, which states that the line integral of a conservative vector field depends only upon the endpoints of the path, and not upon the particular path taken. It is an "iff" theorem, which means "if and only if", which is another way of saying the theorem involves a two-way implication. In other words, if the line integral of a vector field is dependent only upon the endpoints of the path, then it must be conservative.
We started by assuming the line integral depends only upon the endpoints of the path, defined a "phi", and proved that the gradient of this "phi" gives the vector field [Transparancies 1 and 2]. Therefore the vector field is conservative. This is half the theorem.
The then proved the other half of the "iff" [Transparancy 3], by assuming the vector field is conservative and then showing that its line integral is path-independent.
An example was worked-out [Transparancy 4], in which we proved a particular vector field is not conservative. This was done by assuming it is conservative and then showing that this leads to a contradiction in the second-order partial derivatives.
Another example was worked-out [Transparancies 4 and 5] in which a vector field was shown to be conservative by finding its potential function.
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