Sec2Sess38

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directional derivatives

the directional deriv of function w is the change in w caused by a movement (step) of length s in the direction of u (1 unit length vector) coming out of the point on a plane p_0. d(w)/d(s)|p_0,u = grad(w)*p_0 dot_product u

example

find dir deriv take gradient of function and apply it to point values. take the dot product of that result with the unit vector. if provided a direction (v) find the unit vector by u = v/|v| where |...| is the determinant

the directional derivative in the same direction as w is the direction of greatest change. that can be found in the following equation when angle = 0 in the cosine function. that greatest change can also be considered the greatest increase in a function. the greatest decrease in the function can be found by -1*grad(w).

d(w)/d(s)|u=grad(w) dot_prod u = |grad(w)||u|cos(@)=|grad(w)|cos(@) where |u| cancels out due to being 1 and @ is the angle

important distinction: upside down triangle is gradient symbol, right side up triangle is change symbol

distance traveled along a directional derivative can be.found by applying the rate of change found with the gradient to the number of movement steps taken at that change rate. time units can be a type of steps taken along a direction.