VIEWS: 0 PAGES: 7 POSTED ON: 10/3/2012
Gradients and Directional Derivatives Chp 15.6 Putting it all together… • Over the past several classes you have learned how to use and take partial derivatives • Today we look at the essential difference between defining slope along a 2D curve and what it means on a 3D surface or curve x2 10e y Example…What’s the slope of f ( x, y ) at (0,1/2)? 1 x2 y 2 What’s wrong with the way the question is posed? What’s the slope along What’s the slope the direction of the x- along the direction of axis? the y-axis? Quick estimate from the contour plot: z (0 4) 4 y (0.5 0) 0.5 z m 8 Look at this in Excel: The Directional Derivative • We need to specify the direction in which the change occurs… • Define, via a slightly modified Newton quotient: Du f ( x, y) f x ( x, y)a f y ( x, y)b • This specifies the change in the direction of the vector u = <a,b> The Gradient • We can write the Directional derivative as: Du f ( x, y, z ) f x , f y , f z u1 , u2 , u3 Gradient of f(x,y,z) f ( x , y , z ) A Key Theorem • Pg 982 – the Directional derivative is maximum when it is in the same direction as the gradient vector! • Example: If your ski begins to slide down a ski slope, it will trace out the gradient for that surface!
Pages to are hidden for
"Gradients and Directional Derivatives - PowerPoint"Please download to view full document