Transcription of Directional derivatives, steepest a ascent, tangent planes ...
1 Directional derivatives, steepestascent, tangent planesMath 131 Multivariate CalculusD Joyce, Spring 2014 Directional a scalar fieldf:Rn RonRn. So far we have only consideredthe partial derivatives in the directions of the instance f xgives the rate of change along aline parallel to thex-axis. What if we want therate of change in a direction which is not parallelto an axis?First, we can identify directions as unit vectors,those vectors whose lengths equal 1. Letube sucha unit vector, u = 1. Then we define thedirec-tional derivativeoffin the directionuas being thelimitDuf(a) = limh 0f(a+hu) f(a) is the rate of change asx ain the directionu. Whenuis the standard unit vectorei, then,as expected, this Directional derivative is theithpartial derivative, that is,Deif(a) =fxi(a).
2 These Directional derivatives are linear combi-nations of the partial derivatives, at least whenfis differentiable. Note that the directionu=(u1,u2,..,un) is a linear combination of the stan-dard unit vectors:u=u1e1+u2e2+ + ,whenfis differentiable,it is well-approximated by the linear functiongthat de-scribes the tangent plane, that is, byg(x) =f(a) +fx1(a)(x1 a1) + +fxn(a)(xn an).Therefore,Duf(a)= limh 0f(a+hu) f(a)h= limh 0g(a+hu) f(a)h= limh 0fx1(a)hu1+fx2(a)hu2+ +fxn(a)hunh=fx1(a)u1+fx2(a)u2+ +fxn(a)unIn other notation, the Directional derivative is thedot product of the gradient and the directionDuf(a) = f(a) uWe can interpret this as saying that the gradient, f(a), has enough information to find the deriva-tive in any gradient f(a) is a vectorin a certain direction.
3 Letube any direction, thatis, any unit vector, and let be the angle betweenthe vectors f(a) andu. Now, we may concludethat the Directional derivativeDuf(a) = f(a) u= f(a) cos since, in general, the dot product of two vectorsbandcisb c= b c cos but in our case,uis a unit vector. But cos isbetween 1 and 1, so the largest the directionalderivativeDuf(a) can be is when is 0, that iswhenuis the direction of the gradient f(a).In other words, the gradient f(a) points in thedirection of the greatest increase off, that is, thedirection of steepest ascent. Of course, the oppo-site direction, f(a), is the direction of the curves of steepest descentfor the ellipsoid4x2+y2+ 4z2= 16 forz we can describe the projections of the curves inthe (x,y)-plane, that s enough.
4 This ellipsoid is thegraph of a functionf:R2 Rgiven byf(x,y) =12 16 4x2 gradient of this function is f=( f x, f y)=( 2x 16 4x2 y2, y2 16 4x2 y2)The curve of steepest descent will be in the oppositedirection, , we re looking for a pathx(t) = (x(t),y(t))whose derivative is f. In other words, we needtwo functionsx(t) andy(t) such thatx (t) =2x 16 4x2 y2,y (t) =y2 16 4x2 is a differential equation with independentvariablet. We can eliminatetfrom the discussionsincedydx=dydt/dxdt= common method to solve differential equationsis separation of variables, which we can use the last equation, we getdyy=dx4xand, then integrating, dyy= dx4x,soln|y|=14ln|x|+C,which gives us, writingAforeC,|y|=A |x|.
5 That describes the curves of steepest descent as afamily of curves parameterized by the real constantA(different from the last constantA)x= can, of course, use gradi-ents to find equations for planes tangent to typical surface inR3is given by an equationf(x,y,z) = is to say, a surface is a level set of a scalar-valued functionf:R3 R. More generally, a typ-ical hypersurface inRn+1is a level set of a functionf:Rn , the gradient f(a) offpoints in the di-rection of the greatest change off, and vectors or-thogonal to f(a) point in directions of 0 changeoff, that is to say, they lie on the tangent way of saying that is that f(a) is a vectornormal to the surface. Ifxis any point inR3, then f(a) (a x) = 0says that the vectora xis orthogonal to f(a),and therefore lies in the tangent plane, and soxisa point on that 2(Continuous, nondifferentiable func-tion).
6 You re familiar with functions of one vari-able that not continuous everywhere. For example,f(x) =|x|is continuous, and it s differentiable ev-erywhere except atx= 0. The left derivative is 1there, but the right derivative is like that can happen for functions of more2than one variable. Consider the functionf(x) = 0ifx=y= 0xy x2+y2otherwiseThis function is continuous everywhere, but it snot differentiable at (x,y) = (0,0). The graphz=f(x,y) has no tangent plane there. There aredirectional derivatives in two directions, namely,along thex-axis the function is constantly 0, so thepartial derivativedfdxis 0; likewise along they-axis,anddfdyis in all other directions, the Directional deriva-tive does not exist.
7 For instance, along the liney=xthe function isf(x,x) =|x|/ 2, which hasno derivative atx= 131 Home Page ~djoyce/ma131/3
