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**Unformatted text preview: **15.6 Directional Derivatives and the Gradient Vector Let z = f ( x,y ) be the surface S . We have considered the slopes of the tangent line to the curves of intersection when we intersected S with the vertical planes y = y and x = x . These slopes were f x ( x ,y ) and f y ( x ,y ), respectively. Notice that these slopes were the instantaneous rates of change for f in the direction i (in the case of y = y ) and in the direction j (in the case x = x ) at the point ( x ,y ). Question: How would we compute the instantaneous rate of change of f in an arbitrary direction u = < a,b > at ( x ,y )? This instantaneous rate of change should measure the slope of the tangent line of the curve of intersection between S and the vertical plane determined by u through the point ( x ,y ,f ( x ,y )). Method: Let P ( x ,y ,f ( x ,y )) be the point on the surface S . Let Q ( x,y,f ( x,y )) be another point on the surface S and on the curve of intersection in the direction of u . Since Q is on the curve of intersection, the vector < x- x ,y- y > is parallel to the vector u = < a,b > . Hence < x- x ,y- y > = h < a,b > x = x + ha and y = y + hb. Thus the slope of the tangent line to the curve of intersection is given by the limit lim h f ( x,y )- f ( x ,y ) h = lim h f ( x + ha,y + hb )- f ( x ,y ) h , if it exists....

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