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This analogy is very, very tight. If state space
is two-dimensional (e.g. spanned by T and V) then
dP at each point in that space can be represented
by non-square matrix (one row and two columns) that
says: give me an input vector (specifying the
direction and magnitude of a step in state space)
and I'll give you a scalar which is the change in
P that results from that step.