13.Partial_derivative_total_differential.pdf
- 1. Chapter 8: Functions of several variables Nguyen Thu Huong Faculty of Mathematics and Informatics Hanoi University of Science and Technology December 15, 2024
- 2. Functions of several variables Content 8. Functions of several variables 8.4 Higher order partial derivatives 8.5 Tangent planes and approximations 8.6 Total differentials
- 3. Content 8. Functions of several variables 8.4 Higher order partial derivatives 8.5 Tangent planes and approximations 8.6 Total differentials
- 4. Functions of several variables Definition Second order partial derivatives of f (x, y) are: f 00 xx = ∂2f ∂x2 = ∂ ∂x ∂f ∂x f 00 yy = ∂2f ∂y2 = ∂ ∂y ∂f ∂y f 00 xy = ∂2f ∂y∂x = ∂ ∂y ∂f ∂x f 00 yx = ∂2f ∂x∂y = ∂ ∂x ∂f ∂y .
- 5. Functions of several variables Example 1 Compute all the second order partial derivatives of the function z = arctan(x2y). 2 Given z = y cos y x . Compute z00 xy and z00 yx . 3 Given the function f (x, y) = xy x2 − y2 x2 + y2 if (x, y) 6= (0, 0), 0 if (x, y) = (0, 0). Compute f 00 xy (0, 0) and f 00 yx (0, 0).
- 6. Functions of several variables Schwarz theorem Theorem Assume that z = f (x, y) has partial derivatives f 00 xy và f 00 yx near M0(x0, y0) which are continuous at M0. Then, f 00 xy (M0) = f 00 yx (M0). Question: higher derivatives of functions of two variables? Schwarz theorem? higher derivatives of functions of three or more variables?
- 7. Content 8. Functions of several variables 8.4 Higher order partial derivatives 8.5 Tangent planes and approximations 8.6 Total differentials
- 8. Functions of several variables Consider a surface S : z = f (x, y), where f has continuous first partial derivatives. P(x0, y0, z0) ∈ S. Let C1, C2 be two curves obtained by intersecting the planes y = y0 and x = x0 with S at P. Let T1 and T2 be the tangent lines to C1 and C2 at P. The plane contains T1 and T2 is called the tangent plane to the surface S at P.
- 9. Functions of several variables C1 : ( y = y0 z = f (x, y0) ⇒ T1 : ( y = y0 z − z0 = f 0 x (x0, y0)(x − x0). A directional vector of the tangent plane − → u1 = (1; 0; f 0 x (P)). Similarly, − → u2 = (0; 1; f 0 y (P)). Hence, a normal vector is − → n = − → u1 × − → u2 = (−f 0 x (P); −f 0 y (P); 1).
- 10. Functions of several variables Tangent plane equation An equation of the tangent plane to the surface S : z = f (x, y) at the point P(x0, y0, z0) is z − z0 = f 0 x (P)(x − x0) + f 0 y (P)(y − y0) Example Write an equation of the tangent plane at M(1; 1, ; 3) of the surface z = x2y + 2xy2.
- 11. Functions of several variables Linear approximation Suppose that the first order partial derivative of f are continuous. Then for (x, y) close to (x0, y0), we have the linear approximation f (x, y) ≈ f (x0, y0) + f 0 x (x0, y0)(x − x0) + f 0 y (x0, y0)(y − y0). Example f (x, y) = x3 − 2xy2. Approximate f (2.01; 1.98). Example Approximate A = 3 p 1, 022 + 1, 983 − 1; B = ln( 3 √ 1, 02 + 4 √ 0, 98 − 1).
- 12. Content 8. Functions of several variables 8.4 Higher order partial derivatives 8.5 Tangent planes and approximations 8.6 Total differentials
- 13. Functions of several variables Total differentials Definition Given f (x, y): D ⊂ R2 → R, M0(x0, y0) ∈ D.If we can express ∆f = f (x0+∆x, y0+∆y)−f (x0, y0) = A∆x+B∆y+ε1∆x+ε2∆y, where the constants A, B depend only on x0, y0, and ε1 → 0, ε2 → 0 as ∆x, ∆y → 0, we say that f (x, y) is differentiable at M0. df (x0, y0) = A∆x + B∆y: the total diffenrential of f at M0. f is said to be differentiable on D if f is differentiable at all M0 ∈ D. Example Is the function z = 2x − y2 differentiable at (1, 0)?
- 14. Functions of several variables Properties If f is differentiable at M0 then f is continuous at M0. Indeed, let ∆x, ∆y → 0, then ∆f = A∆x + B∆y + ε1∆x + ε2∆y → 0. ∆y = 0, f (x0 + ∆x, y0) − f (x0, y0) ∆x = A + ε1. Passing to the limit as ∆x → 0: A = f 0 x (x0, y0). Similarly, B = f 0 y (x0, y0). df = f 0 x ∆x + f 0 y ∆y. If f is differentiable at M0 then f has partial derivatives at M0. However, the converse is not necessary true. Take f (x, y) = x, then df = ∆x = dx. Similarly, ∆y = dy, df = f 0 x dx + f 0 y dy
- 15. Functions of several variables Example Is f (x, y) = xy x2 + y2 if (x, y) 6= (0, 0), 0 if (x, y) = (0, 0), differentiable at (0, 0)? f is not differentiable at (0, 0) although f 0 x (0, 0) = f 0 y (0, 0) = 0.
- 16. Functions of several variables Theorem (Sufficient condition for differentiability) If f 0 x (x, y), f 0 y (x, y) exist in B(M0, ε) and are continuous at M0. Then, f (x, y) is differentiable at M0 and df (M0) = f 0 x (M0)dx + f 0 y (M0)dy.
- 17. Functions of several variables Example Compute the total differential of the following functions a) z = 1 2 (x2 + y2 ) b) z = xy c) z = arctan xy d) u = z p x2 + y2 , du(1, 2, 3). Recall: linear approximation ∆f = f (x0+∆x, y0+∆y)−f (x0, y0) ≈ f 0 x (x0, y0)∆x + f 0 y (x0, y0)∆y | {z } df (x0,y0) Homework: Textbook page 116: Ex. 3c, 4, 5,7.