Derivatives in Multi
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1) The document discusses the generalization of derivatives from single-variable calculus to multivariable calculus. In multivariable calculus, a function f(x,y) is differentiable at a point (a,b) if the graph near that point is indistinguishable from a plane. 2) The derivative of a function of two variables f(x,y) at a point (a,b) is represented by the Jacobian matrix Jf(a,b), which can be written as the gradient vector grad f(a,b). 3) The directional derivative of f(x,y) at (a,b) in the direction of a unit vector u is
Derivatives in Multi
- 1. DERIVATIVES IN MULTI PORAMATE (TOM) PRANAYANUNTANA In Calc 1: For f : D ⊂ R → R ⊂ R,(1) f (a) := lim x→a f(x) − f(a) x − a .(2) A function f is differentiable at a if, upon continued magnification of the graph about the point (a, f(a)), the graph is indistinguishable from a straight line; that is, f is differentiable at a if lim x→a f(x) − f(a) − f (a)(x − a) x − a = 0.(3) This can be generalized to Multi as follows: In Multi: For f : D ⊂ R2 → R ⊂ R,(4) A function of 2 variables, f(x, y), is differentiable at a point (a, b) if the graph of z = f(x, y) near that point (a, b) is indistinguishable from a plane; that is lim (x, y) r →(a, b) a |f(x, y) − f(a, b) − T( x y − a b ) T(r − a) | r − a = 0,(5) Date: June 20, 2015.
- 2. Derivatives in Multi Poramate (Tom) Pranayanuntana or, from an equation of the tangent plane z = L(x, y) = f(a, b)+fx(a, b)(x−a)+fy(a, b)(y−b) (if it uniquely exists), we have f(x, y) is differentiable at a point (a, b) if lim (x, y) r →(a, b) a |f(x, y) − L(x,y) (f(a, b) + fx(a, b)(x − a) + fy(a, b)(y − b)) | x y − a b = 0.(6) L(x, y) = f(a, b) + fx(a, b) fy(a, b) Jf(a)=Jf(a,b) x − a y − b T(r−a) (7) = f(a, b) + fx(a, b) fy(a, b) grad f(a,b)= f(a,b) x − a y − b , where (◦) = ∂(◦)/∂x ∂(◦)/∂y = (◦)x (◦)y is a vector derivative operator. Compare the following: Calc 1 Multi lim x→a |f(x) − l(x): tangent line of f at a (f(a) + f (a)(x − a)) | |x − a| = 0 lim r→a |f(r) − L(r): tangent plane of f at a (f(a) + Jf(a)(r − a)) | r − a = 0 We can see that Jf(a) is the derivative of z = f(r) = f(x, y) at r = a = (a, b). In this class, we will use dot product instead of matrix multiplication, so the derivative matrix Jf(a) will be represented by the gradient vector, grad f(a, b) = fx(a, b) fy(a, b) , in the xy-plane, which the domain of f is part of. June 20, 2015 Page 2 of 5
- 3. Derivatives in Multi Poramate (Tom) Pranayanuntana Directional Derivatives Calc 1 ∆y = f(x) − f(a) ≈ f (a)(x − a) = f (a)∆x (8) = (f (a) · ˆu) |∆x| = Dˆuf(a) |∆x| , where ˆu is the unit vector pointing in the di- rection of ∆x = x − a. Multi ∆z = f(r) − f(a) ≈ Jf(a)(r − a)(9) = (grad f(a, b) (r − a)) = (grad f(a, b) ˆu) r − a = Dˆuf(a, b) r − a , where ˆu is the unit vector pointing in the direction of ∆r = r − a. The directional derivative of f(x, y) at (a, b) in the direction of ˆu = ∆r ∆r in the domain of f in the xy-plane, denoted by Dˆuf(a, b), is lim Run→0 Rise Run = lim ∆r →0 ∆z ∆r = f(a, b) ∆r ∆r = ( f(a, b) ˆu). June 20, 2015 Page 3 of 5
- 4. Derivatives in Multi Poramate (Tom) Pranayanuntana It can also be seen that Dˆuf(a, b) := lim Run=h→0 Rise f(a+hu1,b+hu2) f( a b + h u1 u2 ) −f(a, b) h Run (10) = lim h=0,h→0 f( x a + hu1, y b + hu2) − f(a, b) h ; with f(x, y) − f(a, b) ≈ fx(a, b)(x − a) + fy(a, b)(y − b) = lim h=0,h→0 fx(a, b)(hu1) + fy(a, b)(hu2) h = fx(a, b) fy(a, b) u1 u2 = ( f(a, b) ˆu) = f(a, b) ˆu cos θ, 0 ≤ θ ≤ π. June 20, 2015 Page 4 of 5
- 5. Derivatives in Multi Poramate (Tom) Pranayanuntana From Dˆuf(a, b) = f(a, b) ˆu cos θ, 0 ≤ θ ≤ π. Since ˆu = 1 and f(a, b) is a fixed positive number (once point (a, b) is picked), therefore max Dˆuf(a, b) = f(a, b) , when cos θ = 1 that is when θ = 0 or when ˆu points in direction of f(a, b). ∆z = f(r) − f(a) ≈ Jf(a)(r − a) = (grad f(a, b) (r − a)) = (grad f(a, b) ˆu) r − a = Dˆuf(a, b) r − a . Properties of grad f(a, b) = f(a, b) If f(a, b) = 0. Then Direction Properties • f(a, b) points in direction of maximum rate of increasing of f(a, b), • Direction of f(a, b) is perpendicular to the contour line z = f(a, b) (in the domain of f in the xy-plane) Magnitude Properties • max Dˆu= f(a,b)/ f(a,b) f(a, b) = f(a, b) 1 ˆu :1 cos 0 That is f(a, b) = max Dˆuf(a, b) or f(a, b) = maximum rate of change of f at (a, b) • f(a, b) is large when contour lines (of fixed ∆z) of f are closer together. is small when contour lines (of fixed ∆z) of f are further apart. June 20, 2015 Page 5 of 5