Lecture 4: Partial and Directional derivatives, Differentiability

Transcription

1 Lecture 4: Partial and Directional derivatives, Differentiability Rafikul Alam Department of Mathematics IIT Guwahati

2 Differential Calculus Task: Extend differential calculus to the functions: Case I: f : A R n R Case II: f : A R R n Case III: f : A R n R m Question: What does it mean to say that f is differentiable?

3 Parametric curve r : R R n A continuous function r : [a, b] R R n is called a parametric curve in R n. The curve Γ := r([a, b]) is parameterized by r(t). Examples: r : R R n given by r(t) := a + tb parameterizes a line in R n passing through a in the direction of b. r : [0, 2π] R 3 given by r(t) := (cos t, sin t, t) parameterizes a circular helix. r : [0, 2π] R 2 given by r(t) := (cos t, sin t) parameterizes the circle x 2 + y 2 = 1.

4 Figure: Line r(t) = p 0 + tv

5 Figure: Helix r(t) = (4 cos t, 4 sin t, t)

6 Figure: Plane curve r(t) = (t 2 sin t, t 2 )

7 Figure: Ellipse r(t) = (6 cos t, 3 sin t)

8 Differentiability of r : R R n Definition: Let r : (a, b) R R n and t 0 (a, b). If r (t 0 ) = dr dt (t 0) := lim t t0 r(t) r(t 0 ) t t 0 exists then r is differentiable at t 0. The derivative r (t 0 ) is called the velocity vector. Fact: r(t) = (r 1 (t),..., r n (t)), where r i : (a, b) R. r is differentiable at t 0 each r i is differentiable at t 0, i = 1, 2,..., n. Further, r (t 0 ) = (r 1(t 0 ),..., r n(t 0 )). r differentiable at t 0 r continuous at t 0.

9 Sum and product rules Fact: Let f, g : (a, b) R R n be differentiable at t 0 (a, b). Then for α R 1. f + g and αf are differentiable at t 0. Further, (f + g) (t) = f (t 0 ) + g (t 0 ) and (αf ) (t 0 ) = αf (t 0 ). 2. f g defined by (f g)(t) := f (t), g(t) is differentiable at t 0 and (f g) (t 0 ) = f (t 0 ) g(t 0 ) + f (t 0 ) g (t 0 ).

10 Velocity and tangent vectors Let r : (a, b) R n be differentiable. Then treating r(t) as the position of a moving object at time t, we have scaled secant = r(t + t) r(t) t r (t) as t 0. But scaled secant tangent vector to the curve at r(t) as t 0. Thus velocity vector v(t) := r (t) is tangent to the curve at r(t). If r(t) := (cos t, sin t) then v(t) = r (t) = ( sin t, cos t).

11 Partial derivatives of f : R 2 R Let f : R 2 R and (a, b) R 2. Then f (a, b) := lim x t 0 f (a + t, b) f (a, b), t when exists, is called partial derivative of f at (a, b) w.r.t to the first variable. Other notations for f (a, b) : x f x (a, b), x f (a, b), 1 f (a, b). Partial derivative f (a, b) w.r.t. the second variable is defined y similarly.

12 Partial derivatives of f : R n R Let f : R n R and a R n. Then f f (a + te i ) f (a) (a) := lim, x i t 0 t when exists, is called partial derivative of f at a w.r.t to the i-th variable. Other notations for f x i (a) : f xi (a), xi f (a), i f (a). If i f (a) exists for i = 1, 2,..., n, then f is said to have first order partial derivatives at a.

13 Examples Consider f : R 2 R given by f (0, 0) := 0 and f (x, y) := xy/(x 2 + y 2 ) for (x, y) (0, 0). Then 1 f (0, 0) = 2 f (0, 0) = 0 even though f is NOT continuous at (0, 0). Consider f : R 2 R given by f (0, 0) = 0 and x sin(1/y) + y sin(1/x) if x 0, y 0, f (x, y) := x sin(1/x) if x 0, y = 0, y sin(1/y) if x = 0, y 0. Then f is continuous at (0, 0) but neither 1 f (0, 0) nor 2 f (0, 0) exists. Moral: Partial derivatives continuity Partial derivatives

14 Sum, product and chain rule Let f, g : R n R and a R n. Suppose i f (a) and i g(a) exist. Then i (αf )(a) = α i f (a) for α R, i (f + g)(a) = i f (a) + i g(a), i (fg)(a) = i f (a)g(a) + f (a) i g(a). If h : R R is differentiable at f (a) then i (h f )(a) exists and i (h f )(a) = h (f (a)) i f (a).

15 Gradient of f : R n R Define φ i : R R by φ i (t) := f (a + te i ). Then i f (a) = φ i (t) φ i (0) lim = φ t 0 t i(0) = d dt f (a + te i) t=0, = rate of change of f at a in the direction e i. Suppose partial derivatives of f : R n R exist at a R n. Then the vector f (a) := ( 1 f (a),..., n f (a)) R n is called the gradient of f at a.

16 Figure: Graph of z = f (x, y) and geometric interpretation of xf (x 0, y 0).

17 Figure: Graph of z = f (x, y) and geometric interpretation of y f (x 0, y 0).

18 Directional derivatives of f : R n R Let f : R n R and a R n. Also let u R n with u = 1. Then the limit, when exists, D u f (a) := f (a + tu) f (a) lim = d t 0 t dt f (a + tu) t=0, = rate of change of f at a in the direction u, is called directional derivative of f at a in the direction u. D u f (a), also denoted by f (a), is the rate of change of f u at a in the direction u.

19 Properties of directional derivatives Let f : R n R and a R n. Also let u R n with u = 1. Then Sum, product and chain rule similar to those of i f (a) hold for D u f (a). If D u f (a) exists for all nonzero u R n then f is said to have directional derivatives in all directions. Obviously i f (a) = D ei f (a). Hence D u f (a) exists in all directions u i f (a) exist for i = 1, 2,..., n.

20 Examples 1. Consider f : R 2 R given by f (x, y) := xy. Then 1 f (0, 0) = 0 = 2 f (0, 0) and f is continuous at (0, 0). However, D u f (0, 0) does NOT exist for u 1 u Consider f : R 2 R given by f (0, 0) = 0 and f (x, y) := x 2 y if (x, y) (0, 0). Then f is NOT x 4 + y 2 continuous at (0, 0), 1 f (0, 0) = 0 = 2 f (0, 0) and D u f (0, 0) exits for all u. Further, D u f (0, 0) = u1/u 2 2 for u 1 u 2 0. Moral: Partial derivatives Directional derivative Continuity Directional derivative. *** End ***