Directional Derivatives PDF

Directional Derivatives Asst. Prof. Saraleen Mae Manongsong Math 251 Differential Geometry Directional Derivatives Directional Derivatives Consider p + tv where p, v ∈ Rn and t ∈ R. For values of t, we can form a straight line p + tv in Rn , and associate a tangent vector vp to this l...

Directional Derivatives Asst. Prof. Saraleen Mae Manongsong Math 251 Differential Geometry Directional Derivatives Directional Derivatives Consider p + tv where p, v ∈ Rn and t ∈ R. For values of t, we can form a straight line p + tv in Rn , and associate a tangent vector vp to this line. Let f be a differentiable function applied on the straight line p + vt, i.e. f (p + tv). The derivative at t = 0, i.e. f (p), gives the initial rate of change of f as p moves in the v direction. Math 251 Differential Geometry Directional Derivatives Directional Derivatives Let f be a differentiable real-valued function on Rn and vp be a tangent vector. The directional derivative of f with respect to vp is given by d vp [f ] = (f (p + tv)) t=0. dt Math 251 Differential Geometry Directional Derivatives Directional Derivatives Let f be a differentiable real-valued function on Rn and vp be a tangent vector. The directional derivative of f with respect to vp is given by d vp [f ] = (f (p + tv)) t=0. dt Example: Let f (x, y, z) = x2 yz be a real-valued function on R3. Clearly, f is differentiable. Determine its directional derivative wrt vp where p = (1, 1, 0) and v = (1, 0, −3). Solution: The straight line associated to vp is given by p + tv = (1, 1, 0) + t(1, 0, −3) = (1 + t, 1, −3t) Evaluating this line at f , we get f (p + tv) = (1 + t)2 · 1 · (−3t) = −3t − 6t2 − 3t3 d Hence dt (f (p + tv)) = −3 − 12t − 9t2 implying that vp [f ] = −3, i.e. f is initially decreasing as p moves in the v direction. Math 251 Differential Geometry Directional Derivatives Directional Derivatives In fact, vp (f ) in terms of the partial derivatives of f at the point p is computed as: n X ∂f vp [f ] = vi (p) i=1 ∂xi where vp = (v1 ,... , vn ). Example: Let f (x, y, z) = x2 yz be a real-valued function on R3. Determine vp [f ] in terms of the partial derivatives of f with vp where p = (1, 1, 0) and v = (1, 0, −3). Solution: We recompute the previous example as follows: ∂f ∂f ∂f = 2xyz, = x2 z, and = x2 y ∂x ∂y ∂z ∂f ∂f ∂f At p = (1, 1, 0), (p) = 0, (p) = 0, and = 1. ∂x ∂y ∂y Hence, vp [f ] = 1 · 0 + 0 · 0 + (−3)1 = −3. Math 251 Differential Geometry Directional Derivatives Activity Let vp be the tangent vector to R3 with v = (2, −1, 3) and p = (2, 0, −1). Compute the directional derivative vp (f ) both using the definition and in terms of the partial derivatives of the following differentiable functions: 1 f = y2 z 2 f = ex cos y Math 251 Differential Geometry Directional Derivatives Directional Derivatives Theorem Let f and g be function on Rn , vp and wp be tangent vectors, and a, b ∈ R. Then, 1 (avp + bwp )[f ] = avp [f ] + bwp [f ] 2 vp [af + bg] = avp [f ] + bvp [g] 3 vp [f g] = vp [f ]g(p) + f (p)vp [g] Q: How do you interpret this theorem in layman’s terms? Math 251 Differential Geometry Directional Derivatives Classroom Activity Let f (x, y, z) = x2 yz and g = y 2 z be a real-valued functions on R3. Exhibit these equalities: 1 (vp + wp )[f ] = vp [f ] + wp [f ] 2 vp [f + g] = vp [f ] + vp [g] 3 vp [f g] = vp [f ]g(p) + f (p)vp [g] where p = (1, 1, 0), v = (1, 0, −3) and w = (0, 1, 1). Show complete solutions. Math 251 Differential Geometry Directional Derivatives Directional Derivatives The operation of a vector field V on a function f is a real-valued function, denoted V [f ], whose value at each point p ∈ Rn is V (p)[f ], that is, the derivative of f wrt the tangent vector V (p) at p. Analogy: In calculus, for a function f on the real line, the derivative df function is dx , whose value at each point p is the derivative of f at df that point, dx (p). ∂f Remark: If Ui is the natural frame field, then Ui [f ] =. ∂xi For instance, when i = 1 so that U1 (p) = (1, 0, 0), ∂f d U1 (p)[f ] = (p) = (f (p1 + t, p2 , p3 )) ∂x1 dt for any p = (p1 , p2 , p3 ) ∈ R3. Math 251 Differential Geometry Directional Derivatives Corollary Let V, W be vector fields on Rn and f, g, h be real-valued function, 1 (f V + gW )[h] = f V [h] + gW [h] 2 V [af + bg] = aV [f ] + bV [g] 3 V [f g] = V [f ]g + f V [g] for a, b ∈ R. Math 251 Differential Geometry Directional Derivatives Directional Derivatives ∂f Recall that if Ui is the natural frame field, then Ui [f ] =. ∂xi ∂f Note: The result Ui [f ] = makes it easy to carry out explicit ∂xi computations. For example, if V = xU1 − y 2 U3 and f = x2 y + z 3 then V [f ] = xU1 [x2 y] + xU1 [z 3 ] − y 2 U3 [x2 y] − y 2 U3 [z 3 ] = x(2xy) + 0 − 0 − y 2 (3z 2 ) = 2x2 y − 3y 2 z 2 Math 251 Differential Geometry Directional Derivatives Classroom Activity Let V = y 2 U1 − xU3 and f = xy, g = z 3. Compute V [f ] and V [f g]. Math 251 Differential Geometry Directional Derivatives Activity Let V = y 2 U1 − xU3 and f = xy, g = z 3. Compute f V [g] − gV [f ] and V [V [f ]]. Math 251 Differential Geometry Directional Derivatives