Two techniques have been devised for the exploration of electrostatic and magnetic fields where conventional methods fail. Both use electron lens systems to magnify the deflecting effect of the fields on electron beams. One technique is the electron optical analog of the ``Schlieren'' method; the other involves deformed shadows of a thin obstruction.For several reasons the ``Schlieren'' patterns obtained experimentally thus far have been interpreted only qualitatively. The patterns obtained by the shadow method, however, have been used for quantitative evaluation of field strengths. Experimental patterns of both types, produced by the fields of magnetized recording wires, are shown.The shadow method is theoretically applicable to the quantitative evaluation of a wide variety of electric and magnetic fields. It utilizes formulas based on relations between field distribution and electron deflection, and between the latter and the geometrical parameters of a focusing system. Such formulas are derived and applied to several simple types of field.An application of the theory to the magnetized wires used in the experiments yields relations from which each field distribution is evaluated by fitting a line to experimentally determined points. The theory also predicts the geometrical properties of the experimental patterns.The agreement between theory and experiment is sufficiently good to justify the underlying theoretical assumptions and approximations.