DOCSLIB.ORG

CE07.32 Seismic Design of Structures

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

9.7 Gradient of a scalar field. Directional derivative Some of the vector fields in applications can be obtained from scalar fields. This is very advantageous because scalar fields can be handled more easily. The gradient of a given scalar function f (,xyz ,) denoted by grad f or f (read nabla f ) is the vector function, f ff grad ff i j k . (9.7.1) x yz Here x,,yz are Cartesian coordinates in a domain in 3-space in which f is defined and differentiable. (The symbol is also called del - pg. 283 Hildebrand.) Example: If f (,xyz ,) 2 y3 4 xz 3 x then gradf (4zy 3)i 62 j 4xk . The differential operator (read nabla or del) is defined by, i j k . (9.7.1*) x yz Gradients are useful in several ways, notably in giving the rate of change of f (,xyz ,) in any direction in space, in obtaining surface normal vectors, and in deriving vector fields from scalar fields. 9.7.1 Directional derivative From calculus we know that ,, give the rates of change of f (,xyz ,) in the direction of x yz the coordinate axes. It seems natural to extend this and ask for the rate of change of f in an arbitrary direction in space. The directional derivative Dfb or df /ds of a function f (,,)xyz at a point P in the direction of a vector b is, df f() Q f () P Dfb lim . (9.7.2) dss0 s Here Q is a variable point on the straight line L in the direction of b , and s is the distance between P and Q. Also s 0 if Q lies in the direction of b (see Fig. 213), s 0 if Q lies in the direction of b . May 30, 2011 9.7-1 Next, use Cartesian xyz -coordinates and for b a unit vector. Then the line L is given by ri()sxsyszs () ()j ()kp 0 s b, (9.7.3) where p0 is the position vector of P. Equation (9.7.2) now shows that Db f df/ ds is the derivative of the function f (xs ( ), ys ( ), zs ( )) with respect to the arc length s of L. Hence, assuming that f has continuous partial derivatives and applying the chain rule df f f f Df x y z, (9.7.4) b ds x y z where primes denotes derivatives with respect to s (which are taken at s 0 ). But here differentiating (9.7.3) gives ri()sxyzj kb . Hence (9.7.4) is simply the inner product of grad f and b , df Dfb grad f. (9.7.5) b ds (Note that in eqn (9.7.5) b 1.) Example 1: Gradient. Directional derivative Find the directional derivative of f (,xyz ,) 2 x22 3 y z2 at P :(2,1,3) in the direction of ai2 k. Solution: gradf 4xyzi 6j 2 k , gives at P the vector gradfP ( ) 8i 6j 6k . 1 baˆ (2 i k). 5 14 DfP() (ik 2)(86 i jk 6) . b 55 The minus sign indicates that at P the function f is decreasing in the direction of a . 9.7.2 Gradient is a vector Since grad f is defined in terms of components depending on the Cartesian coordinates, to prove that grad f is actually a vector, we must show that grad f has a length and direction independent f ff of the choice of those coordinates. In contrast i 2 jk also looks like a vector but does x yz not have a length and direction independent of the choice of Cartesian coordinates. grad f points in the direction of maximum increase of f . Theorem 1: Vector character of gradient. Maximum increase Let f ()(,,)Pfxyz be a scalar function having continuous first partial derivatives in some domain B in space. Then grad f exists in B and is a vector (i.e., its length and direction are independent of the particular choice of Cartesian coordinates). If gradfP ( ) 0 at some point P, it has the direction of maximum increase of f at P . May 30, 2011 9.7-2 Proof: From eqn (9.7.5), Dfb bb grad f grad f cos grad f cos , (9.7.6) where is the angle between b and grad f . Now f is a scalar function. Hence its value at a point P depends on P but not on the particular choice of coordinates. The same holds for the arc length s of the line L (Fig. 213), hence also for Dfb . Now (9.7.6) shows that Dfb is maximum when 0 and then Dfb grad f. It follows that the length and direction of grad f are independent of the choice of coordinates. Since 0 if and only if b has the direction of grad f , the latter is the direction of maximum increase of f at P , provided grad f 0 at P . 9.7.3 Gradient as surface normal vector Let S be a surface represented by f (,,)x y z c const , where f is differentiable. Such a surface is called a level surface of f , and for different c we get different level surfaces. Let C be a curve on S through a point P of S . As a curve in space, C has a representation ri()txtytzt() ()j ()k . For C to lie on the surface S , the components of r()t must satisfy f (,xy,)z c, i.e., f ((),(),())xt yt zt c. (9.7.7) A tangent vector of C is ri()txtytzt () ()j ()k (pg. 392 EK). The tangent vectors of all curves on S passing through P will generally form a plane, called the tangent plane of S at P . (Exceptions occur at edges or cusps of S , for instance, for the cone in Fig. 215 at the apex.) The normal of this plane (the straight line through P perpendicular to the tangent plane) is called the surface normal to S at P . Differentiating eqn (9.7.7) with respect to t , fff xyzgrad f r 0 . xyz Hence, grad f is orthogonal to all the vectors r in the tangent plane, so that it is a normal vector of S at P . Theorem 2: Gradient as surface normal vector Let f be a differentiable scalar function in space. Let f (,,)x y z c const represent a surface S . Then if the gradient of f at a point P of S is not the zero vector, it is a normal vector of S at P . May 30, 2011 9.7-3 Example 2: Gradient as surface normal vector. Cone Find a unit normal vector n of the cone of revolution zxy224(2 ) at the point P :(1,0,2). Solution: The cone is the level surface f 0 of f (,xyz ,) 4( x22 y ) z2. Thus 1 gradf 8xyzi 8j 2 k , gradfP () 8ik 4 . nik (2 ) . 5 The vector n points downward since it has a negative z -component. The other unit normal vector of the cone at P is n . 9.7.4 Vector fields that are gradients of scalar fields ("Potentials") Some vector fields have the advantage that they can be obtained from scalar fields, which can be handled more easily. Such a vector field is given by a vector function v()P which is obtained as the gradient of a scalar function, v()Pf grad()P. The function f ()P is called a potential function or a potential of v()P . Such a v()P and the corresponding vector field are called conservative because in such a vector field, energy is conserved; i.e., no energy is lost (or gained) in displacing a body (or a charge in the case of an electric field) from a point P to another point in the field and back to P . (See Sec. 10.2.) Conservative fields play a central role in physics and engineering. A basic application concerns the gravitational force (Ex. 3 in Sec. 9.4) and we show that it has a potential which satisfies Laplace's equation, the most important partial differential equation in physics and its applications. May 30, 2011 9.7-4 Theorem 3: Gravitational field. Laplace's equation The force of attraction, cc pr [(xx ) i ( yy )j ( zz )k ], (9.7.8) rr33000 between two particles at points Px0000:( ,yz , ) and Pxyz:(,,) (as given by Newton's law of gravitation) has the potential f (,xyz ,) c/ r, where r (0) is the distance between P0 and P . c Thus p gradf grad . This potential f is a solution of Laplace's equation, r 222fff 2 f 0 . (9.7.9) xyz222 (2 f (read nabla squared f , or del squared f ) is called the Laplacian of f .) 2221/2 Proof: The distance is rxxyyzz[()(000 )( )]. ccc pipp123j pki grad f j k x ryrzr dr1 cx() x0 dr1 cy() y0 pc1 3 , pc2 3 , dr r x r dr r y r dr1 cz() z0 pc3 3 dr r z r p given by eqn (9.7.8) is the gradient of the scalar function (,,)/f xyz c r. Consider 2 ff cx() x 1 3() x x 2 00c . 233 5 xxxx r r r Similarly 2222 ff113(yy00 ) 3(zz ) 235cc , 235 . yrr zrr 222fff 0 . Hence 2 f 0 . xyz222 2 f is also denoted by f . The differential operator 222 2 , (9.7.11) x22yz2 read "nabla squared" or "del squared" or delta, is called the Laplace operator. It can be shown that the field of force produced by any distribution of masses is given by a vector function that is the gradient of a scalar function f and f satisfies eqn.
Recommended publications
  • The Directional Derivative the Derivative of a Real Valued Function (Scalar field) with Respect to a Vector

    The Directional Derivative the Derivative of a Real Valued Function (Scalar field) with Respect to a Vector

    Math 253 The Directional Derivative The derivative of a real valued function (scalar field) with respect to a vector. f(x + h) f(x) What is the vector space analog to the usual derivative in one variable? f 0(x) = lim − ? h 0 h ! x -2 0 2 5 Suppose f is a real valued function (a mapping f : Rn R). ! (e.g. f(x; y) = x2 y2). − 0 Unlike the case with plane figures, functions grow in a variety of z ways at each point on a surface (or n{dimensional structure). We'll be interested in examining the way (f) changes as we move -5 from a point X~ to a nearby point in a particular direction. -2 0 y 2 Figure 1: f(x; y) = x2 y2 − The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point . z -5 -2 0 y 2 Figure 2: f(x; y) = x2 y2 − y If we give the direction by using a second vector, say ~u, then for →u X→ + h any real number h, the vector X~ + h~u represents a change in X→ position from X~ along a line through X~ parallel to ~u. →u →u h x Figure 3: Change in position along line parallel to ~u The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point .
  • Notes for Math 136: Review of Calculus

    Notes for Math 136: Review of Calculus

    Notes for Math 136: Review of Calculus Gyu Eun Lee These notes were written for my personal use for teaching purposes for the course Math 136 running in the Spring of 2016 at UCLA. They are also intended for use as a general calculus reference for this course. In these notes we will briefly review the results of the calculus of several variables most frequently used in partial differential equations. The selection of topics is inspired by the list of topics given by Strauss at the end of section 1.1, but we will not cover everything. Certain topics are covered in the Appendix to the textbook, and we will not deal with them here. The results of single-variable calculus are assumed to be familiar. Practicality, not rigor, is the aim. This document will be updated throughout the quarter as we encounter material involving additional topics. We will focus on functions of two real variables, u = u(x;t). Most definitions and results listed will have generalizations to higher numbers of variables, and we hope the generalizations will be reasonably straightforward to the reader if they become necessary. Occasionally (notably for the chain rule in its many forms) it will be convenient to work with functions of an arbitrary number of variables. We will be rather loose about the issues of differentiability and integrability: unless otherwise stated, we will assume all derivatives and integrals exist, and if we require it we will assume derivatives are continuous up to the required order. (This is also generally the assumption in Strauss.) 1 Differential calculus of several variables 1.1 Partial derivatives Given a scalar function u = u(x;t) of two variables, we may hold one of the variables constant and regard it as a function of one variable only: vx(t) = u(x;t) = wt(x): Then the partial derivatives of u with respect of x and with respect to t are defined as the familiar derivatives from single variable calculus: ¶u dv ¶u dw = x ; = t : ¶t dt ¶x dx c 2016, Gyu Eun Lee.
  • Tensor Calculus and Differential Geometry

    Tensor Calculus and Differential Geometry

    Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane .
  • Matrix Calculus

    Matrix Calculus

    Appendix D Matrix Calculus From too much study, and from extreme passion, cometh madnesse. Isaac Newton [205, §5] − D.1 Gradient, Directional derivative, Taylor series D.1.1 Gradients Gradient of a differentiable real function f(x) : RK R with respect to its vector argument is defined uniquely in terms of partial derivatives→ ∂f(x) ∂x1 ∂f(x) , ∂x2 RK f(x) . (2053) ∇ . ∈ . ∂f(x) ∂xK while the second-order gradient of the twice differentiable real function with respect to its vector argument is traditionally called the Hessian; 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂x1 ∂x1∂x2 ··· ∂x1∂xK 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 2 K f(x) , ∂x2∂x1 ∂x2 ··· ∂x2∂xK S (2054) ∇ . ∈ . .. 2 2 2 ∂ f(x) ∂ f(x) ∂ f(x) 2 ∂xK ∂x1 ∂xK ∂x2 ∂x ··· K interpreted ∂f(x) ∂f(x) 2 ∂ ∂ 2 ∂ f(x) ∂x1 ∂x2 ∂ f(x) = = = (2055) ∂x1∂x2 ³∂x2 ´ ³∂x1 ´ ∂x2∂x1 Dattorro, Convex Optimization Euclidean Distance Geometry, Mεβoo, 2005, v2020.02.29. 599 600 APPENDIX D. MATRIX CALCULUS The gradient of vector-valued function v(x) : R RN on real domain is a row vector → v(x) , ∂v1(x) ∂v2(x) ∂vN (x) RN (2056) ∇ ∂x ∂x ··· ∂x ∈ h i while the second-order gradient is 2 2 2 2 , ∂ v1(x) ∂ v2(x) ∂ vN (x) RN v(x) 2 2 2 (2057) ∇ ∂x ∂x ··· ∂x ∈ h i Gradient of vector-valued function h(x) : RK RN on vector domain is → ∂h1(x) ∂h2(x) ∂hN (x) ∂x1 ∂x1 ··· ∂x1 ∂h1(x) ∂h2(x) ∂hN (x) h(x) , ∂x2 ∂x2 ··· ∂x2 ∇ .
  • Math 1320-9 Notes of 3/30/20 11.6 Directional Derivatives and Gradients

    Math 1320-9 Notes of 3/30/20 11.6 Directional Derivatives and Gradients

    Math 1320-9 Notes of 3/30/20 11.6 Directional Derivatives and Gradients • Recall our definitions: f(x0 + h; y0) − f(x0; y0) fx(x0; y0) = lim h−!0 h f(x0; y0 + h) − f(x0; y0) and fy(x0; y0) = lim h−!0 h • You can think of these definitions in several ways, e.g., − You keep all but one variable fixed, and differentiate with respect to the one special variable. − You restrict the function to a line whose direction vector is one of the standard basis vectors, and differentiate that restriction with respect to the corresponding variable. • In the case of fx we move in the direction of < 1; 0 >, and in the case of fy, we move in the direction for < 0; 1 >. Math 1320-9 Notes of 3/30/20 page 1 How about doing the same thing in the direction of some other unit vector u =< a; b >; say. • We define: The directional derivative of f at (x0; y0) in the direc- tion of the (unit) vector u is f(x0 + ha; y0 + hb) − f(x0; y0) Duf(x0; y0) = lim h−!0 h (if this limit exists). • Thus we restrict the function f to the line (x0; y0) + tu, think of it as a function g(t) = f(x0 + ta; y0 + tb); and compute g0(t). • But, by the chain rule, d f(x + ta; y + tb) = f (x ; y )a + f (x ; y )b dt 0 0 x 0 0 y 0 0 =< fx(x0; y0); fy(x0; y0) > · < a; b > • Thus we can compute the directional derivatives by the formula Duf(x0; y0) =< fx(x0; y0); fy(x0; y0) > · < a; b > : • Of course, the partial derivatives @=@x and @=@y are just directional derivatives in the directions i =< 0; 1 > and j =< 0; 1 >, respectively.
  • Derivatives Along Vectors and Directional Derivatives

    Derivatives Along Vectors and Directional Derivatives

    DERIVATIVES ALONG VECTORS AND DIRECTIONAL DERIVATIVES Math 225 Derivatives Along Vectors Suppose that f is a function of two variables, that is, f : R2 → R, or, if we are thinking without coordinates, f : E2 → R. The function f could be the distance to some point or curve, the altitude function for some landscape, or temperature (assumed to be static, i.e., not changing with time). Let P ∈ E2, and assume some disk centered at p is contained in the domain of f, that is, assume that P is an interior point of the domain of f.This allows us to move in any direction from P , at least a little, and stay in the domain of f. We want to ask how fast f(X) changes as X moves away from P , and to express it as some kind of derivative. The answer clearly depends on which direction you go. If f is not constant near P ,then f(X) increases in some directions, and decreases in others. If we move along a level curve of f,thenf(X) doesn’t change at all (that’s what a level curve means—a curve on which the function is constant). The answer also depends on how fast you go. Suppose that f measures temperature, and that some particle is moving along a path through P . The particle experiences a change of temperature. This happens not because the temperature is a function of time, but rather because of the particle’s motion. Now suppose that a second particle moves along the same path in the same direction, but faster.
  • Directional Derivatives

    Directional Derivatives

    Directional Derivatives The Question Suppose that you leave the point (a, b) moving with velocity ~v = v , v . Suppose further h 1 2i that the temperature at (x, y) is f(x, y). Then what rate of change of temperature do you feel? The Answers Let’s set the beginning of time, t = 0, to the time at which you leave (a, b). Then at time t you are at (a + v1t, b + v2t) and feel the temperature f(a + v1t, b + v2t). So the change in temperature between time 0 and time t is f(a + v1t, b + v2t) f(a, b), the average rate of change of temperature, −f(a+v1t,b+v2t) f(a,b) per unit time, between time 0 and time t is t − and the instantaneous rate of f(a+v1t,b+v2t) f(a,b) change of temperature per unit time as you leave (a, b) is limt 0 − . We apply → t the approximation f(a + ∆x, b + ∆y) f(a, b) fx(a, b) ∆x + fy(a, b) ∆y − ≈ with ∆x = v t and ∆y = v t. In the limit as t 0, the approximation becomes exact and we have 1 2 → f(a+v1t,b+v2t) f(a,b) fx(a,b) v1t+fy (a,b) v2t lim − = lim t 0 t t 0 t → → = fx(a, b) v1 + fy(a, b) v2 = fx(a, b),fy(a, b) v , v h i·h 1 2i The vector fx(a, b),fy(a, b) is denoted ~ f(a, b) and is called “the gradient of the function h i ∇ f at the point (a, b)”.
  • Lecture Notes of Thomas' Calculus Section 11.5 Directional Derivatives

    Lecture Notes of Thomas' Calculus Section 11.5 Directional Derivatives

    Lecture Notes of Thomas’ Calculus Section 11.5 Directional Derivatives, Gradient Vectors and Tangent Planes Instructor: Dr. ZHANG Zhengru Lecture Notes of Section 11.5 Page 1 OUTLINE • Directional Derivatives in the Plane • Interpretation of the Directional Derivative • Calculation • Properties of Directional Derivatives • Gradients and Tangents to Level Curves • Algebra Rules for Gradients • Increments and Distances • Functions of Three Variables • Tangent Panes and Normal Lines • Planes Tangent to a Surface z = f(x, y) • Other Applications Why introduce the directional derivatives? • Let’s start from the derivative of single variable functions. Consider y = f(x), its ′ ′ derivative f (x0) implies the rate of change of f. f (x0) > 0 means f increasing as x ′ becomes large. f (x0) < 0 means f decreasing as x becomes large. • Consider two-variable function f(x, y). The partial derivative fx(x0,y0) implies the rate of change in the direction of~i while keep y as a constant y0. fx(x0,y0) > 0 implies increasing in x direction. Another interpretation, the vertical plane passes through (x0,y0) and parallel to x-axis intersects the surface f(x, y) in the curve C. fx(x0,y0) means the slope of the curve C. Similarly, The partial derivative fy(x0,y0) implies the rate of change in the direction of ~j while keep x as a constant x0. • It is natural to consider the rate of change of f(x, y) in any direction ~u instead of ~i and ~j. Therefore, directional derivatives should be introduced. ... Lecture Notes of Section 11.5 Page 2 Definition 1 The derivative of f at P0(x0,y0) in the direction of the unit vector ~u = u1~i + u2~j is the number df f(x0 + su1,y0 + su2) − f(x0,y0) = lim , ds s→0 s ~u,P0 provided the limit exists.
  • Multivariable Calculus Review

    Multivariable Calculus Review

    Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multivariable Calculus Review Multivariable Calculus Review Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multivariable Calculus Review n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0. n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if Multivariable Calculus Review n I ν(αx) = jαjν(x) 8 x 2 R α 2 R n I ν(x + y) ≤ ν(x) + ν(y) 8 x; y 2 R We usually denote ν(x) by kxk. Norms are convex functions. lp norms 1 Pn p p kxkp := ( i=1 jxi j ) ; 1 ≤ p < 1 kxk1 = maxi=1;:::;n jxi j Outline Multi-Variable Calculus Point-Set Topology Compactness The Weierstrass Extreme Value Theorem Operator and Matrix Norms Mean Value Theorem Multi-Variable Calculus Norms: n n A function ν : R ! R is a vector norm on R if n I ν(x) ≥ 0 8 x 2 R with equality iff x = 0.
  • Derivatives in N-Dimensional Spaces

    Derivatives in N-Dimensional Spaces

    5 DERIVATIVES IN N-DIMENSIONAL VECTOR SPACES A Introduction , -' ,.- * -..: The approach of our textbook (as d&ioped in the previous two units) in emphasizing the geometrical aspects of partial derivatives, directional derivatives, and the gradient is, in many ways, an excellent approach. We get a picture of what appears to be happening, and, especially for the beginning student, this is much more meaningful than a more precise and general discussion that would apply to all vector spaces, regardless of dimension. In fact, if we want to use the latter approach, it is wise to use the former first. This is in accord with many such decisions which have already been made .in our previous treatment of mathematics. As an example, within the framework of our present course, notice how carefully we ex- ploited the geometric aspects of vectors as arrows before we even mentjoned n-dimensional vector spaces. And, in this respect, recall how sophisticated many of our "arrow" results were in their own right. That is,.granted that arrows were simpler than n-tuples, when all we had were arrows, there were still some rather difficult ideas for us to assimilate. In any event, our aim in this chapter is to continue within the spirit of the game of mathematics which, thus far, has served us so well. Namely, we have introduced n-dimensional vector spaces so that we could utilize the similarity in form between the previously studied functions of the form f(x) and our new functions of the form f(x). For example, when it came time to define continuity and limits for functions of several real variables, we rewrote f(xl, ...,xn) as f (5) and then mimicked the f (x) situation by copying definitions verbatim, subject only to making the appropriate "vectorizations." Rather than review this procedure, let us instead move to the next plateau.
  • Generalized Directional Derivatives and Subgradients of Nonconvex

    Generalized Directional Derivatives and Subgradients of Nonconvex

    ( )i,\ , . t, ., \\\rt to.o j -i .1.. GENERT\LIZED DIRECTIO\AL DERI\TATIVES AND SUBGRADIENTS OF NONCONVEX FUNCTIONS R T. ROCKAFIJLLAI{ l lntroduction. Srudjes of optimjzation prollens and ccrlah kiD(ls of difierentiaL eqLrations hale led in rcccnt )enrs to thc dclelopnrent of a gen cralized rheor] of dift-erentiaiion qLrite distincr iD spirit ancl rarse of applicati,m from the onc base.l on L. Schwari:z's 'distriblriions. ' This thco( associares .l-r,,t-\. u" t, r. .. o ., ;r . oto,og: .r, .. ,. L point r a I certain elements of the clLral space E+ caLLed \tLy tints tit gtarlitnt js stneralircd s ot .f at '. 1 lcsc form A set dI(r) rhat a]lrat.s conre: and Ncak*-closed L'possiLl_v cmprl ). The nrlrifurctior df : r + bflx) is thc Rules that relate Dl to scncraljzed djrectional cterilatives of /, or:rlloN d,i 1o b€ expr.ssed or estimarcd ;n terns of thc sLrbcljlTerenrjaLs of other tunctjoos (\i hen.l : ,h + f'. f = ! o ,1, etc ), co prise the vrrd.r'fl./.rrill ar/.&/,s. Such rrLes arc used espcciall! ir anall,zing ihe con.Liijon 0 a D.f(s), Nhich i]picalt,v Deans that:f is some sort of'quasi-optlnaL'poinr ior,l The cxtcnded-real- laluccl nrtlrc of,i is essential in such a contcxr as a delicc for representins SulijlTerenrial caiclils bcgan rrith com-cx lLrrctions on R". Rockafeilar l43l defincd d,f ior su.h funcriorrs, shon'cd hor t.} characicrjze A_ft]j) nr rerms of one sidcd cLirectional rleriratiles _l'l:;r r), and prolecl that rndcr mild restrictions rLlles sllcI .s (1.1) d(.n +/,)(r) : d,no) + dj,ir) are valj.l.
  • Derivation of the Formula for a Directional Derivative

    Derivation of the Formula for a Directional Derivative

    Derivation of the Directional Derivative Formula We let ~u = u1~{ + u2~| be a fixed unit vector in the xy-plane, and as usual, we let z = f(x; y). We want to show that the simple formula for the directional derivative f~u = fxu1 + fyu2 is valid wherever f is a differentiable function. We fix a particular point (a; b) in the xy-plane where f is differentiable, and we recall that the formal definition of fu(a; b) is f(a + u1h; b + u2h) f(a; b) fu(a; b) = lim − ; h 0 h ! so we want to show that the limit simplifies to fx(a; b)u1 + fy(a; b)u2 : ( ) ∗ Now we can certainly rewrite the difference quotient in the definition in the following form, dividing it into one part where the x-value varies while y remains fixed and the other where the reverse holds: f(a + u1h; b + u2h) f(a; b) f(a + u1h; b + u2h) f(a; b + u2h) f(a; b + u2h) f(a; b) − = − u1 + − u2 : ( ) h u1h u2h ∗∗ Because the change in the x-value in the first term is u1h, we have arranged for the denominator to have that value, in hopes that it will approach the partial derivative of f with respect to x; and similarly for y in the second term. Now as h 0, we also have u2h 0, so the fraction [f(a; b + u2h) f(a; b)]=(u2h) ! ! − in the second term goes to fy(a; b). And the factors u1 and u2 are constants.