As we asked on Friday about irrotational fields and conservative fields: is the converse true? That is, if the field is solenoidal, can it necessarily be written as the curl of some vector potential, A? It turns out again that the converse is only partially true, as given by a theorem which holds, as before, only for simply connected domains.
This theorem was not proven in the lecture, but we attempted to understand, qualitatively, what the vector potential means. The flow lines of a vector potential A are best visualized as being "wrapped around" the flow lines of the vector field F. A diagram was shown [Transparancy 2] of a constant vector field (the divergence of a constant vector field is automatically zero) and its vector potential wrapped around it.
It was noted that a vector potential is somewhat arbitrary in that we have the freedom of adding the gradient of any scalar field to it. (The curl of any gradient is zero.) Another diagram was shown [Transparancy 2] of the previous constant vector field, in which a gradient added to the vector potential gave the vector potential a spiralling "downstream" component. (This also happens to be the diagram shown on the front cover of the textbook!) The actual vector field and its physical interpretation, of course, remains the same.
This invariance of physics under an arbitrary change of gradient is called "gauge invariance". It is important in electromagnetic theory and also in particle physics where "Gauge Field Theories" play an important role.
Oriented Surfaces were introduced [Transparancy 3]. Positive directions are defined in terms of the Right Hand Rule. However, some surfaces (eg. a Mobius strip) are non-orientable.The parametric representation of a surface was given [Transparancy 4]. Tangent vectors were obtained in terms of the two parameters which describe the surface, and a normal was obtained from the cross product of these tangent vectors. A surface element was then defined [Transparancy 5] in terms of a double integral over the magnitude of the cross product of the tangent vectors.
An example was considered [Transparancies 6 and 7] in which we looked at a conical surface. The surface is best parameterized in terms of spherical coordinates with fixed phi. The tangent vectors were obtained, from which we found the normal vector and the surface element. This surface element agrees with what one would obtain from the usual "surface of revolution" method. The vector method, however, has the advantage of generalizing to more complex surfaces which can't be described as a surface of revolution.
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