Math 1320-9 Notes of 3/30/20 11.6 Directional Derivatives and Gradients • Recall our definitions: f(x0 + h; y0) − f(x0; y0) fx(x0; y0) = lim h−!0 h f(x0; y0 + h) − f(x0; y0) and fy(x0; y0) = lim h−!0 h • You can think of these definitions in several ways, e.g., − You keep all but one variable fixed, and differentiate with respect to the one special variable. − You restrict the function to a line whose direction vector is one of the standard basis vectors, and differentiate that restriction with respect to the corresponding variable. • In the case of fx we move in the direction of < 1; 0 >, and in the case of fy, we move in the direction for < 0; 1 >. Math 1320-9 Notes of 3/30/20 page 1 How about doing the same thing in the direction of some other unit vector u =< a; b >; say. • We define: The directional derivative of f at (x0; y0) in the direc- tion of the (unit) vector u is f(x0 + ha; y0 + hb) − f(x0; y0) Duf(x0; y0) = lim h−!0 h (if this limit exists). • Thus we restrict the function f to the line (x0; y0) + tu, think of it as a function g(t) = f(x0 + ta; y0 + tb); and compute g0(t). • But, by the chain rule, d f(x + ta; y + tb) = f (x ; y )a + f (x ; y )b dt 0 0 x 0 0 y 0 0 =< fx(x0; y0); fy(x0; y0) > · < a; b > • Thus we can compute the directional derivatives by the formula Duf(x0; y0) =< fx(x0; y0); fy(x0; y0) > · < a; b > : • Of course, the partial derivatives @=@x and @=@y are just directional derivatives in the directions i =< 0; 1 > and j =< 0; 1 >, respectively. • In the plane, any unit vector can be written as < cos θ; sin θ >. Math 1320-9 Notes of 3/30/20 page 2 • Example 2: Find the directional derivative Duf(x; y) if f(x; y) = x3 − 3xy + 4y2 p < 3;1> where u = 2 is the unit vector corresponding to the angle θ = π=6. What is Duf(1; 2)? Math 1320-9 Notes of 3/30/20 page 3 • The vector @f @f rf(x; y) =< f (x; y); f (x; y) >= i + j x y @x @y is called the gradient of f at (x; y). • Note: The gradient is an example of a vector function of a vector argument. The calculus of such functions is called Vector Calculus. It is covered in Chapter 13 of our textbook, but is beyond our scope this semester. • With the notion of the gradient we can rewrite our formula for directional derivatives as Duf = rf · u Math 1320-9 Notes of 3/30/20 page 4 • Example 4: Find the directional derivative of the function f(x; y) = x2y3 − 4y at the point (2; −1) in the direction of the vector v = 2i + 5j Math 1320-9 Notes of 3/30/20 page 5 Functions of Three Variables • Everything carries over. The directional derivative is the dot product of the gradient and the unit direction vector. The gradient of f = f(x; y; z) is @f @f @f rf(x; y; z) = i + j + k =< f ; f ; f > @x @y @z x y z • Same thing in n variables. • Example 5: For f(x; y; z) = x sin yz find the gradient of f and find the directional derivative of f at (1; 3; 0) in the direction of the vector < 1; 2; −1 >. Math 1320-9 Notes of 3/30/20 page 6 • Major issue: What direction maximizes the directional derivative? What direction minimizes it? How does the direction of the gradient relate to the direction of contour lines? Math 1320-9 Notes of 3/30/20 page 7 Tangent Planes to Level Surfaces • For functions of three variables the gradient is orthogonal to level surfaces. • Suppose we have a surface defined by F (x; y; z) = C = F (x0; y0; z0) for some point (x0; y0; z0) on the level surface. Then the gradient is orthogonal to the tangent plane the the level surface through that point. • Example V: Find the equations of the tangent plane and normal line at the point (−2; 1; −3) to the ellipsoid x2 z2 + y2 + = 3: 4 9 Math 1320-9 Notes of 3/30/20 page 8 Math 1320-9 Notes of 3/30/20 page 9 Summary • A directional derivative is the dot product of the gradient and a unit direction vector. • You can also think of a directional derivative as an ordi- nary derivative of the restriction of a function of several variables to a line. • The gradient is the direction of steepest ascent. • The negative gradient is the direction of steepest descent. • The gradient is orthogonal to level curves and level sets. • Partial derivatives are just directional derivatives in the directions of the coordinate axes. Math 1320-9 Notes of 3/30/20 page 10.