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Gradients and Directional Derivatives R Horan & M Lavelle Intermediate Mathematics Gradients and Directional Derivatives R Horan & M Lavelle The aim of this package is to provide a short self assessment programme for students who want to obtain an ability in vector calculus to calculate gradients and directional derivatives. Copyright c 2004 [email protected] , [email protected] Last Revision Date: August 24, 2004 Version 1.0 Table of Contents 1. Introduction (Vectors) 2. Gradient (Grad) 3. Directional Derivatives 4. Final Quiz Solutions to Exercises Solutions to Quizzes The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials. Section 1: Introduction (Vectors) 3 1. Introduction (Vectors) The base vectors in two dimensional Cartesian coordinates are the unit vector i in the positive direction of the x axis and the unit vector j in the y direction. See Diagram 1. (In three dimensions we also require k, the unit vector in the z direction.) The position vector of a point P (x, y) in two dimensions is xi + yj . We will often denote this important vector by r . See Diagram 2. (In three dimensions the position vector is r = xi + yj + zk .) y 6 Diagram 1 y 6 Diagram 2 P (x, y) 3 r 6 j yj 6- - - - 0 i x 0 xi x Section 1: Introduction (Vectors) 4 The vector differential operator ∇ , called “del” or “nabla”, is defined in three dimensions to be: ∂ ∂ ∂ ∇ = i + j + k . ∂x ∂y ∂z Note that these are partial derivatives! This vector operator may be applied to (differentiable) scalar func- tions (scalar fields) and the result is a special case of a vector field, called a gradient vector field. Here are two warming up exercises on partial differentiation. ∂ Quiz Select the following partial derivative, (xyzx). ∂z 2 x−1 x−1 (a) x yz , (b) 0 , (c) xy logx(z) , (d) yz . ∂ Quiz Choose the partial derivative (x cos(y) + y). ∂x (a) cos(y) , (b) cos(y) − x sin(y) + 1 , (c) cos(y) + x sin(y) + 1 , (d) − sin(y) . Section 2: Gradient (Grad) 5 2. Gradient (Grad) The gradient of a function, f(x, y), in two dimensions is defined as: ∂f ∂f gradf(x, y) = ∇f(x, y) = i + j . ∂x ∂y The gradient of a function is a vector field. It is obtained by applying the vector operator ∇ to the scalar function f(x, y) . Such a vector field is called a gradient (or conservative) vector field. Example 1 The gradient of the function f(x, y) = x + y2 is given by: ∂f ∂f ∇f(x, y) = i + j ∂x ∂y ∂ ∂ = (x + y2)i + (x + y2)j ∂x ∂y = (1 + 0)i + (0 + 2y)j = i + 2yj . Section 2: Gradient (Grad) 6 Quiz Choose the gradient of f(x, y) = x2y3. (a) 2xi + 3y2j , (b) x2i + y3j , (c) 2xy3i + 3x2y2j , (d) y3i + x2j . The definition of the gradient may be extended to functions defined in three dimensions, f(x, y, z): ∂f ∂f ∂f ∇f(x, y) = i + j + k . ∂x ∂y ∂z Exercise 1. Calculate the gradient of the following functions (click on the green letters for the solutions). (a) f(x, y) = x + 3y2 , (b) f(x, y) = px2 + y2 , √ 1 (c) f(x, y, z) = 3x2 y + cos(3z) , (d) f(x, y, z) = , px2 + y2 + z2 4y (e) f(x, y) = , (f) f(x, y, z) = sin(x)ey ln(z) . (x2 + 1) Section 3: Directional Derivatives 7 3. Directional Derivatives To interpret the gradient of a scalar field ∂f ∂f ∂f ∇f(x, y, z) = i + j + k , ∂x ∂y ∂z note that its component in the i direction is the partial derivative of f with respect to x. This is the rate of change of f in the x direction since y and z are kept constant. In general, the component of ∇f in any direction is the rate of change of f in that direction. Example 2 Consider the scalar field f(x, y) = 3x + 3 in two dimen- sions. It has no y dependence and it is linear in x. Its gradient is given by ∂ ∂ ∇f = (3x + 3)i + (3x + 3)j ∂x ∂y = 3i + 0j . As would be expected the gradient has zero component in the y di- rection and its component in the x direction is constant (3). Section 3: Directional Derivatives 8 Quiz Select a point from the answers below at which the scalar field f(x, y, z) = x2yz − xy2z decreases in the y direction. (a) (1, −1, 2) , (b) (1, 1, 1) , (c) (−1, 1, 2) , (d) (1, 0, 1) . Definition: if nˆ is a unit vector, then nˆ ·∇f is called the directional derivative of f in the direction nˆ. The directional derivative is the rate of change of f in the direction nˆ. Example 3 Let us find the directional derivative of f(x, y, ) = x2yz in the direction 4i − 3k at the point (1, −1, 1). √ The vector 4i−3k has magnitude p42 + (−3)2 = 25 = 5. The unit 1 vector in the direction 4i − 3k is thus nˆ = 5 (4i − 3k). The gradient of f is ∂ ∂ ∂ ∇f = (x2yz)i + (x2yz)j + (x2yz)k ∂x ∂y ∂z = 2xyzi + x2zj + x2yk , Section 3: Directional Derivatives 9 and so the required directional derivative is 1 nˆ · ∇f = (4i − 3k) · (2xyzi + x2zj + x2yk) 5 1 = 4 × 2xyz + 0 − 3 × x2y . 5 At the point (1, −1, 1) the desired directional derivative is thus 1 nˆ · ∇f = [8 × (−1) − 3 × (−1)] = −1 . 5 Exercise 2. Calculate the directional derivative of the following func- tions in the given directions and at the stated points (click on the green letters for the solutions). (a) f = 3x2 − 3y2 in the direction j at (1,2,3). (b) f = px2 + y2 in the direction 2i + 2j + k at (0,−2,1). (c) f = sin(x) + cos(y) + sin(z) in the direction πi + πj at (π,0,π ). Section 3: Directional Derivatives 10 We now state, without proof, two useful properties of the direc- tional derivative and gradient. • The maximal directional derivative of the scalar field f(x, y, z) is in the direction of the gradient vector ∇f. • If a surface is given by f(x, y, z) = c where c is a constant, then the normals to the surface are the vectors ±∇f. Example 4 Consider the surface xy3 = z +2. To find its unit normal at (1, 1, −1) , we need to write it as f = xy3 − z = 2 and calculate the gradient of f: ∇f = y3i + 3xy2j − k . At the point (1, 1, −1) this is ∇f = i + 3j − k. The magnitude of this maximal rate of change is √ p12 + 32 + (−1)2 = 11 . 1 Thus the unit normals to the surface are ±√ (i + 3j − k). 11 Section 3: Directional Derivatives 11 Quiz Which of the following vectors is normal to the surface x2yz = 1 at (1,1,1)? (a) 4i + j + 17k , (b) 2i + j + 2k , (c) i + j + k , (d) −2i − j − k . Quiz Which of the following vectors is a unit normal to the surface cos(x)yz = −1 at (π,1,1)? 1 1 2 (a) −√ j + √ k , (b) πi + j + √ k , 2 2 π 1 1 (c) i , (d) −√ j − √ k . 2 2 Quiz Select a unit normal to the (spherically symmetric) surface x2 + y2 + z2 = 169 at (5,0,12). 1 1 1 1 1 (a) i + j − k , (b) i + j + k , 6 6 3 3 3 5 12 5 12 (c) i + k , (d) − i + k . 13 13 13 13 Section 4: Final Quiz 12 4. Final Quiz Begin Quiz Choose the solutions from the options given. 1. What is the gradient of f(x, y, z) = xyz−1? y x xy (a) i + j − z−2k , (b) i + j − k , z z z2 1 (c) yz−1i + xz−1j + xyz−2k , (d) − . z2 2. If n is a constant, choose the gradient of f(r) = 1/rn, where r = |r| and r = xi + yj + zk. n i + j + k nr n r (a) 0 , (b) − , (c) − , (d) − . 2 rn+1 rn+2 2 rn+2 3. Select the unit normals to the surface of revolution, z = 2x2 +2y2 at the point (1,1,4). 1 1 (a) ±√ (i + j − k) , (b) ±√ (i + j + k) , 3 3 1 1 (c) ±√ (i + j) , (d) ±√ (2i + 2j − 4k) . 2 2 End Quiz Solutions to Exercises 13 Solutions to Exercises Exercise 1(a) The function f(x, y) = x + 3y2 , has gradient ∂f ∂f ∇f(x, y) = i + j ∂x ∂y ∂ ∂ = (x + 3y2)i + (x + 3y2)j ∂x ∂y = (1 + 0)i + (0 + 3 × 2y2−1)j = i + 6yj . Click on the green square to return Solutions to Exercises 14 Exercise 1(b) The gradient of the function p 2 2 1 f(x, y) = x2 + y2 = (x + y ) 2 is given by: ∂f ∂f ∂ 2 2 1 ∂ 2 2 1 ∇f(x, y) = i + j = (x + y ) 2 i + (x + y ) 2 j ∂x ∂y ∂x ∂y 1 2 2 1 −1 ∂ 2 = (x + y ) 2 × (x )i 2 ∂x 1 2 2 1 −1 ∂ 2 + (x + y ) 2 × (y )j 2 ∂y 1 2 2 − 1 2−1 1 2 2 − 1 2−1 = (x + y ) 2 × 2x i + (x + y ) 2 × 2y j 2 2 2 2 − 1 2 2 − 1 = (x + y ) 2 xi + (x + y ) 2 yj x y = i + j . px2 + y2 px2 + y2 Click on the green square to return Solutions to Exercises 15 Exercise 1(c) The gradient of the function 2√ 2 1 f(x, y, z) = 3x y + cos(3z) = 3x y 2 + cos(3z) , is given by: ∂f ∂f ∂f ∇f(x, y, z) = i + j + k ∂x ∂y ∂z 1 ∂ 2 2 ∂ 1 ∂ = 3y 2 (x ) i + 3x (y 2 ) j + (cos(3z)) k ∂x ∂y ∂z 1 2−1 2 1 1 −1 = 3y 2 × 2x i + 3x × y 2 j − 3 sin(3z) k 2 1 3 2 − 1 = 6y 2 x i + x y 2 j − 3 sin(3z) k 2 √ 3 x2 = 6x y i + √ j − 3 sin(3z) k .
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