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Home Vector Calculus Directional Derivatives and Gradients The Unit Vector  
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The Unit Vector
The partial derivatives ∂z/∂x and ∂z/∂y are the rates of change of z = f(x, y) as the point (x, y) moves in the direction of the xaxis and the yaxis. We now consider the rate of change of z as the point (x, y) moves in other directions. Let P(a, b) be a point in the (x, y) plane and let U = cos αi + sin αj be a unit vector, a is the angle from the xaxis to U (see Figure 13.1.1). The line through P with direction vector U has the vector equation X = P + tU or in parametric form, (1) x = a + t cos α, Figure 13.1.1 The unit vector At t = 0 we have x = a and y = b. If we intersect the surface z = f(x, y) with the vertical plane through the line (Equation 1), we obtain the curve z = f(a + t cos α, b + t sin α) = F(t). The slope dz/dt = F'(0) of this curve at f = 0 is called the slope or derivative of f in the U direction and is written f_{U}(a, b) (Figure 13.1.2). Figure 13.1.2: The directional derivative Here is the precise definition. DEFINITION Given a function z = f(x, y) and a unit vector U = cos αi + sin αj, the derivative of f in the U direction is the limit f_{U}(a, b) is called a directional derivative of f at (a, b). The partial derivatives of f(x, y) are equal to the derivatives of f(x, y) in the i and j directions:


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