euler's formula

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Definition of euler's formula in English English dictionary: Formula which links complex exponentiation with trigonometric functions:
e^{i \theta} = \cos \theta + i \sin \theta.; Formula which calculates the normal curvature of an arbitrary direction in the tangent plane in terms of the principal curvatures \kappa_1 and \kappa_2 and the angle \theta which that direction makes with the first principal direction:
\kappa_n(\theta) = \kappa_1 \cos^2 \theta + \kappa_2 \sin^2 \theta.; Either of two important mathematical theorems of Leonhard Euler. The first is a topological invariance (see topology) relating the number of faces, vertices, and edges of any polyhedron. It is written F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges. A cube, for example, has 6 faces, 8 vertices, and 12 edges, and satisfies this formula. The second formula, used in trigonometry, says e^ix = cos x + isin x where e is the base of the natural logarithm and i is the square root of -1 (see irrational number). When x is equal to or 2, the formula yields two elegant expressions relating , e, and i: e^i = -1 and e^2i = 1