Sec12.5 - Sec.12.5 Directional Derivatives and the Gradient...

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Sec.12.5 Directional Derivatives and the Gradient Vector A directional derivative allows us to find the rate of change (slope) of a function in two or more variables in the direction of any unit vector u = < a, b > = cos ,sin θ < > . (A partial derivative is in direction of unit vectors i and j.) Let surface S have equation z = f(x,y) and z 0 = f(x 0 ,y 0 ) be point P on surface S. The vertical plane passing through P in the direction of u at point P is the rate of change of z in the direction u. If Q(x,y,z) is another point on C and P and Q are projections of P and Q on the xy plane, then P Q is parallel to u. So () , P Q hu ha hb ′′ == < > for some h . u b=sin a=cos (x 0 , y 0 ) y x
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The directional derivative of f at (x 0 ,y 0 ) in the direction of unit vector u = < a, b > is 00 0 0 0 (, ) ( , ) (, )l i m u h f xh a yh b f x y Df x y h ++ = if this limit exist. Theorem: If f is a differentiable function of x and y , then f has a directional derivative in the direction of any unit vector u = < a, b > and ( ) ,( , ) ( , ) ux y Df xy f xya f xyb =+ .
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Sec12.5 - Sec.12.5 Directional Derivatives and the Gradient...

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