This paper shows how the radon transform can be used to determine vector fields. A scheme to determine the velocity field of a moving fluid by measurements with a continuous doppler signal is suggested. When the flow is confined to a bounded domain, as is the case in most applications, it can be uniquely decomposed into one gradiental and one rotational part. The former vanishes if the fluid is incompressible and source-free, and the latter can be completely reconstructed by the methods proposed in this paper if the domain is simply connected. Special attention is paid to laminar flow in a long cylindrical vessel with circular cross-section. Under such conditions the flow profile becomes parabolic, which makes the vessel recognizable as a typical `N-shaped` pattern in an image describing the rotation of the velocity field. The vessel yields the same doppler tomographic pattern, no matter how it is sectioned. The ideas presented should be applicable also when studying the flow in blood vessels, even if the flow profile in these is not quite parabolic. The discrepancies only make the `N-shape` somewhat distorted.