Vectors Calaulus 1.13

The Gradient Directional Derivatives Suppose you are given a topographical map and want to see how steep it is from a point that is  neither due West or due North.  Recall that the slopes due north and due west are the two partial  derivatives.  The slopes in other directions will be called the directional derivatives.  Formally, we define           Definition   Let f(x,y) be a differentiable function and let u be a unit vector then the directional derivative of f in the direction of u is        Note that if u is i then the directional derivative is just fx and if u is j the it is fy. Just as there is a difficult and an easy way to compute partial derivatives, there is a difficult way  and an easy way to compute directional derivatives.           Theorem Let f(x,y) be a differentiable function, and u be a unit vector with direction q, then Example: Let         f(x,y)  =  2x + 3y2 - xy and         v  =  <3,2> Find         Dv f(x,y) Solution We have          fx  =  2 - y and          fy  =  6y - x and         Hence         Dv f(x,y)  =  <2 - y, 6y - x> . <3/, 2/>                   2                         3              =                (2 - y)  +            (6y - x)                                       Exercise Let         Find  Dv f(x,y) The Gradient We define   grad f  =  x, fy> Notice that                Du f(x,y) = (grad f) . u The gradient has a special place among directional derivatives.  The theorem below states this  relationship.                Theorem If grad f(x,y) = 0 then for all u,  Du f(x,y) = 0 The direction of grad f(x,y) is the direction with maximal directional derivative. The direction of -grad f(x,y) is the direction with the minimal directional derivative. Proof: If         gradf(x,y)  =  0 then           Du f(x,y)  =  grad f . u  =  0 . u  =  0 Du f(x,y)  =  grad f . u  =  ||grad f || cos q This is a maximum when q = 0 and a minimum when q = p.  If q = 0 then grad f and u  point in the same direction.  If q = p then u and grad f point in opposite directions.  This proves 2 and 3. Example:   Suppose that a hill has altitude         w(x,y)  =  x2 - y Find the direction that is the steepest uphill and the steepest downhill at the point (2,3). Solution We find         grad w  =  <2x, -y>  =  <4, -3> Hence the steepest uphill is in the direction         <4,-3> while the steepest downhill is in the direction         -<4,-3>  =  <-4,3> The Gradient and Level Curves If f is differentiable at (a,b) and grad f is nonzero at (a,b) then grad f is perpendicular to the level curve through (a,b).