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Directional Derivatives and Gradients (Three Variables)
Directional derivatives and gradients for functions of three variables are similar to the case of two variables. DEFINITION Given a real function w = f(x, y, z) and a unit vector U = cos αi + cos βj + cos γk in space, the derivative of f in the direction U and the gradient of f at (a, b, c) are defined as follows. THEOREM 2 Suppose w = f(x, y, z) is smooth at (a, b, c). Then for any unit vector U = cos αi + cos βj + cos γk, the directional derivative f_{U}(a, b, c) exists and Corollaries 1 and 2 also hold for functions of three variables. In Corollary 2, grad f is normal to the tangent plane of the level surface f(x, y, z)  f(a, b, c) = 0 at (a, b, c).


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