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Chap14_Sec6

# Chap14_Sec6 - PARTIAL DERIVATIVES 14.6 Directional...

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PARTIAL DERIVATIVES 14.6 Directional Derivatives nd the Gradient Vector and the Gradient Vector In this section, we will learn how to find: The rate of changes of a function of two or more variables in any direction.

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DIRECTIONAL DERIVATIVES Recall that, if z = f ( x , y ), then the partial derivatives f x and f y are defined as: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ( , ) ( , ) ( , ) lim ( , ) ( , ) ( , ) lim x h y h f x h y f x y f x y h f x y h f x y f x y h + - = + - = They represent the rates of change of z in the x - and y -directions—that is, in the directions of the unit vectors i and j . Suppose that we now wish to find the rate of change of z at ( x 0 , y 0 ) in the direction of an arbitrary unit vector u = < a , b > .
DIRECTIONAL DERIVATIVES To do this, we consider the surface S with equation z = f ( x , y ) [the graph of f ] and we let z 0 = f ( x 0 , y 0 ). s Then, the point P ( x 0 , y 0 , z 0 ) lies on S . The vertical plane that passes through P in the direction of u intersects S in a curve C . The slope of the tangent line T to C at the point P is the rate of change of z in the direction of u . Now, let: s Q ( x , y , z ) be another point on C. s P’ , Q’ be the projections of P , Q on the xy -plane.

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DIRECTIONAL DERIVATIVES Then, the vector is parallel to u . So, for some scalar h . Therefore,
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Chap14_Sec6 - PARTIAL DERIVATIVES 14.6 Directional...

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